Limits and Continuity

Introduction

Limits provide the mathematical language for describing what happens to a quantity as another quantity approaches a specified value. They are central to calculus because derivatives, integrals, differential-equation models, numerical algorithms, and feedback-system analyses are all built on limiting ideas. In mechatronics, limits help engineers analyze sensor behavior near operating thresholds, actuator saturation, robot trajectories, signal transitions, and the validity of mathematical models near singular configurations.

A limit is not necessarily the value of a function at a point. Instead, it describes the value that a function approaches as its input gets arbitrarily close to that point. This distinction is essential when a function is undefined at the point, when the function has a removable discontinuity, or when the behavior differs from the left and right.

Learning Objectives

After studying this chapter, you should be able to:

1. Interpret limits graphically, numerically, and algebraically.
2. Evaluate limits using direct substitution, algebraic simplification, conjugates, and standard limit laws.
3. Distinguish two-sided, one-sided, infinite, and limits at infinity.
4. Identify vertical, horizontal, and oblique asymptotes.
5. Recognize and resolve common indeterminate forms.
6. Define continuity and classify discontinuities.
7. Apply limits and continuity to engineering models and system response.
8. Assess the limitations of numerical evidence and idealized models.

1. The Meaning of a Limit

Let f(x) be defined for values of x near a, although it may or may not be defined at a. We write

when the values of f(x) approach L as x approaches a from either direction. The variable x need not equal a in this statement.

For example, consider

Factoring gives f(x) = (x − 1)(x + 1)/(x − 1) = x + 1 for x ≠ 1. Therefore,

The original expression is undefined at x = 1, but its limit exists. The graph has a missing point, or hole, at (1, 2).

1.1 Numerical interpretation

A numerical estimate can be obtained by evaluating the function at values on both sides of the target point.

The values approach 2 from both sides. Numerical investigation is useful for forming a conjecture, but it does not by itself prove a limit. Rounding error, insufficiently close sample points, and hidden oscillations can produce misleading evidence.

1.2 Formal ε–δ definition

The rigorous definition states that

if, for every ε > 0, there exists δ > 0 such that

Here ε is the permitted output error and δ is the required input distance. This definition formalizes the idea that the output can be made as close to L as desired by restricting x sufficiently close to a, without requiring x = a.

2. Evaluating Limits Algebraically

2.1 Direct substitution

If f is continuous at a, then

Polynomials are continuous for every real input. Rational functions are continuous wherever their denominators are nonzero. Standard functions such as sin x, cos x, e^x, and ln x are continuous throughout their domains.

Example 1: Polynomial limit

Evaluate lim_x→2(3x³ − 4x + 1). Direct substitution gives

Thus the limit is 17.

Example 2: Rational limit with a nonzero denominator

Direct substitution is valid because the denominator approaches 4, not zero.

2.2 Limit laws

Suppose lim_x→af(x) = L and lim_x→ag(x) = M, where the limits exist. Then:

For a positive integer n,

These laws follow from the algebraic properties of real numbers and allow complicated expressions to be decomposed into simpler parts.

2.3 Factoring and cancellation

When direct substitution produces 0/0, the expression is an indeterminate form, not an answer. Algebraic simplification may reveal the limiting behavior.

Example 3: Factoring

Factor the numerator:

Therefore,

Cancellation is valid for nearby values with x ≠ 2. It does not assert that the original function is defined at 2.

2.4 Rationalization using a conjugate

Radical expressions often become manageable after multiplication by a conjugate.

Example 4: A square-root limit

Evaluate

Multiplying by the conjugate gives

Hence the limit is

2.5 Standard trigonometric limits

Angles must be measured in radians for these results:

A useful consequence is

where k is a constant and x is measured in radians.

Example 5

The radian condition is important. If angles are expressed in degrees, the numerical value of the corresponding ratio changes.

2.6 The squeeze theorem

If

for all x sufficiently close to a, and

then

For example, since −1 ≤ cos(1/x) ≤ 1, multiplying by x² ≥ 0 gives

Both bounding functions approach zero as x → 0, so

The factor cos(1/x) oscillates increasingly rapidly, but its bounded amplitude is suppressed by x².

3. One-Sided Limits and Discontinuity

The left-hand limit is

and describes values as x approaches a from values less than a. The right-hand limit is

and describes values from values greater than a. A two-sided limit exists exactly when both one-sided limits exist and are equal:

Example 6: Jump discontinuity

Let

Then

whereas

Because the one-sided limits differ, the two-sided limit does not exist. This is a jump discontinuity.

One-sided analysis is particularly important for piecewise control laws, unilateral mechanical contact, diode models, saturation logic, and state-dependent switching.

4. Infinite Limits and Asymptotes

An infinite limit describes unbounded growth rather than convergence to a finite number. For example,

This notation means that the function becomes arbitrarily large as x approaches zero from the right; it does not mean that infinity is an ordinary real number.

If a function becomes unbounded near x = a, then x = a is a vertical asymptote. For

x = 3 is a vertical asymptote. The signs differ on either side:

4.1 Limits at infinity

A limit at infinity describes long-range behavior:

If this limit is finite, y = L is a horizontal asymptote.

For a rational function, compare the degrees of the numerator and denominator:

Example 7: Horizontal asymptote

Dividing numerator and denominator by x² makes this clear:

Thus y = 2 is a horizontal asymptote.

Example 8: Oblique asymptote

Polynomial division gives

Since 2/(x−1) → 0 as x → ±∞, the oblique asymptote is y = x + 1.

Asymptotes describe trends, not necessarily forbidden crossings. A graph may cross a horizontal or oblique asymptote.

5. Indeterminate Forms

The expressions 0/0, ∞/∞, 0·∞, ∞−∞, 1^∞, 0⁰, and ∞⁰ are indeterminate forms. They indicate that further analysis is needed; they do not determine a unique limit.

For example, each of the following has form 0/0 at x = 0, but a different limit:

Appropriate methods include factoring, rationalization, common denominators, standard limits, substitution, and—when its hypotheses are satisfied—L’Hôpital’s rule.

5.1 L’Hôpital’s rule

If f and g are differentiable on an interval near a, g′(x) ≠ 0, and the quotient has form 0/0 or ∞/∞, then, under the relevant existence conditions,

provided the limit on the right exists as a finite value or as an appropriate infinite limit.

Example 9

This is 0/0. Differentiating numerator and denominator gives

L’Hôpital’s rule should not be applied merely because a quotient is present. The required indeterminate form and differentiability conditions must first be checked.

5.2 Exponential indeterminate forms

For positive f(x), logarithms can transform powers. If y = f(x)^g(x), then

One evaluates the limit of ln y and then exponentiates. This method is useful for forms such as 1^∞.

6. Continuity

A function f is continuous at x = a if all three conditions hold:

1. f(a) is defined.
2. lim_x→af(x) exists.
3. lim_x→af(x) = f(a).

Continuity means that sufficiently small changes in input produce sufficiently small changes in output near the point. It does not mean that the function is globally smooth, differentiable, or free of rapid variation.

A function is continuous on an interval if it is continuous at every point in that interval. At an endpoint, one-sided continuity is used. For instance, continuity from the right at a requires lim_x→a⁺f(x) = f(a).

6.1 Types of discontinuity

6.2 Continuity of piecewise functions

Consider

To make f continuous at 2, equate the left-hand limit and the value from the second branch:

and

Continuity requires 2k − 1 = 5, so k = 3.

6.3 The Intermediate Value Theorem

If f is continuous on the closed interval [a,b], and N lies between f(a) and f(b), then there is at least one c ∈ [a,b] such that

For a zero, if f(a) and f(b) have opposite signs, continuity guarantees at least one root between them. This theorem justifies bracketing methods such as bisection. It guarantees existence, not uniqueness, and does not by itself provide the root’s numerical value.

7. Numerical Limits and Engineering Computation

In practical engineering, limits are often investigated through sampled data, simulation, symbolic algebra, or numerical solvers. A table of values can reveal trends, while a plot can expose discontinuities, saturation, and asymptotic behavior. However, finite precision imposes limitations.

Near a removable discontinuity, direct evaluation may cause numerical failure even though the limit exists. For example, evaluating (x²−1)/(x−1) for a floating-point value extremely close to 1 can suffer from subtractive cancellation. Rewriting it as x + 1 is both analytically valid for nearby unequal inputs and numerically more stable.

Best practices include:

• evaluate from both sides when a two-sided limit is claimed;
• reduce algebraic cancellation before numerical evaluation;
• use radians in trigonometric limit computations;
• inspect scale, units, and significant figures;
• distinguish a sampled approximation from a proof;
• test sensitivity to step size and floating-point precision;
• verify that a model’s domain matches the physical system.

A numerical result such as f(1.000001) ≈ 2.000001 is evidence about one input, not a complete statement about lim_x→1f(x).

8. Application to Mechatronic Systems

Limits and continuity connect mathematical models with physical response. Suppose a sensor output is modeled by

where θ is angular position in radians, V₀ is an offset voltage in volts, and K is sensitivity in volts per radian. This model is continuous and predicts a smooth output for every angle within its operating range. A discontinuity in measured data may indicate switching, quantization, saturation, loose wiring, or an invalid model rather than a mathematical curiosity.

A common actuator approximation is saturation:

Here r is the requested command and U_max is the maximum allowed command, with both measured in consistent actuator units. The function is continuous at the boundaries if the middle branch reaches ±U_max, but its derivative changes abruptly there. Thus continuity alone does not imply differentiability.

In feedback systems, a sudden change in a control law can create sharp command transitions, increased mechanical stress, or excitation of unmodeled dynamics. Engineers may use smoothing functions, rate limits, hysteresis, or carefully designed switching logic. These modifications must be evaluated against response speed, safety, stability, and actuator constraints.

9. Illustrative Case: Proximity-Sensor Model Near a Blind Zone

Background

Consider an illustrative ultrasonic proximity sensor mounted on a mobile robot. A simplified calibration model predicts distance from a measured echo time. The model is reliable only outside a blind zone caused by transducer recovery.

Problem

Suppose the estimated distance is modeled by

where t is echo time in seconds and c is the speed of sound in metres per second. The factor 2 accounts for the outgoing and returning path. At room temperature, an approximate value is c = 343 m/s, but the actual value depends on temperature, humidity, and air composition.

The sensor rejects measurements for t < t_b, where t_b = 0.00020 s. At the threshold, an engineer wants to know whether the accepted measurement model joins continuously with a reported minimum distance of 0.0343 m.

Analysis

The model’s threshold distance is

The units are consistent because seconds cancel, leaving metres. Define the complete reporting rule as

The left-hand limit at the threshold is 0.0343 m, and the right-hand limit is also

The assigned value is 0.0343 m, so the reporting function is continuous at t_b. Nevertheless, the derivative changes: the left branch has slope 0 m/s with respect to time, while the right branch has slope 343/2 = 171.5 m/s. The output is continuous but has a corner.

Approach and outcome

The continuity check indicates that no idealized distance jump occurs at the threshold. However, the corner and the physical blind zone can still produce undesirable robot behavior if a controller interprets rapidly changing measurements as precise motion information. A robust implementation should include validity flags, filtering appropriate to the control bandwidth, timeout handling, and conservative obstacle-clearance logic.

Learning points

1. A piecewise engineering model must be checked at every joining point.
2. Continuity prevents an idealized jump but does not guarantee a smooth derivative.
3. Units provide an essential check on model correctness.
4. A mathematical limit does not remove sensor noise, quantization, latency, or environmental dependence.
5. Safety decisions should not rely on extrapolation beyond the validated operating domain.

10. Challenges, Limitations, and Professional Practice

A mathematical model may be continuous while the physical system is not. Mechanical backlash, impacts, digital sampling, sensor quantization, communication delays, and discrete logic can produce behavior that differs from an ideal continuous model. Conversely, a measured jump may be an artifact of thresholding rather than a true physical discontinuity.

Model validity must therefore be considered alongside limit calculations. Engineers should identify the domain of calibration, state assumptions explicitly, document units, and test behavior near boundaries. In safety-related systems, threshold choices should be traceable and verified through simulation, hardware-in-the-loop testing, and controlled experiments. Data-driven or AI-based models also require boundary testing: a smooth interpolated prediction does not establish physically valid behavior outside the training data.

Continuity can support existence arguments and numerical methods, but it does not guarantee stability, boundedness, differentiability, or uniqueness. These properties require additional analysis. For example, a continuous control input can still destabilize a plant, and a continuous nonlinear equation can have multiple solutions.

Chapter Summary

A limit describes the value approached by a function as its input approaches a specified point, independently of whether the function is defined at that point. Limits can be evaluated by direct substitution, limit laws, factoring, rationalization, standard trigonometric limits, the squeeze theorem, and—when its conditions apply—L’Hôpital’s rule. One-sided limits determine whether a two-sided limit exists. Infinite limits describe unbounded behavior and identify vertical asymptotes, while limits at infinity identify horizontal or oblique asymptotes.

Indeterminate forms require further analysis rather than immediate numerical interpretation. Continuity at a point requires that the function be defined there, that the limit exist, and that the limit equal the function value. Discontinuities may be removable, jump, infinite, or oscillatory. The Intermediate Value Theorem provides an existence guarantee for roots and supports bracketing methods.

In mechatronics, limits and continuity help engineers interpret sensor thresholds, actuator saturation, switching laws, asymptotic system behavior, and model boundaries. Numerical evidence is valuable but must be distinguished from proof and checked against precision, units, assumptions, and physical validity.

Key Terms

Limit: The value a function approaches as its input approaches a specified value.

Two-sided limit: A limit approached from both the left and the right.

Left-hand limit: The limit as the input approaches from values less than the target.

Right-hand limit: The limit as the input approaches from values greater than the target.

Continuity: The property that a function is defined, has a limit, and equals that limit at a point.

Removable discontinuity: A discontinuity where a finite limiting value exists but is missing or incorrectly assigned.

Jump discontinuity: A discontinuity caused by unequal one-sided limits.

Infinite limit: A limit in which function values become arbitrarily large in magnitude.

Vertical asymptote: A line x = a near which a function becomes unbounded.

Horizontal asymptote: A line y = L approached as the input tends to positive or negative infinity.

Oblique asymptote: A nonhorizontal linear trend approached at infinity.

Indeterminate form: An expression such as 0/0 that requires further analysis.

Squeeze theorem: A theorem that determines a limit by bounding a function between two functions with the same limit.

L’Hôpital’s rule: A derivative-based method for certain quotient indeterminate forms.

ε–δ definition: The formal definition relating an output tolerance to an input tolerance.

Intermediate Value Theorem: A theorem guaranteeing intermediate function values for continuous functions.

Asymptotic behavior: The long-range or near-boundary trend of a function.

Piecewise function: A function defined by different formulas on different portions of its domain.

Model validity: The extent to which a mathematical model accurately represents a physical system within stated conditions.

Review Questions

Short Answer Questions

1. Explain why lim_x→af(x) does not require f(a) to be defined.
2. State the three conditions required for continuity at a point.
3. What must be true of the one-sided limits for a two-sided limit to exist?
4. Distinguish a vertical asymptote from a removable discontinuity.
5. Why must angles be measured in radians when using lim_x→0(sin x)/x = 1?

Long Answer Questions

1. Explain and compare direct substitution, factoring, rationalization, and L’Hôpital’s rule as methods for evaluating limits. Include conditions for their use.
2. Describe the principal types of discontinuity and give a mathematical or engineering example of each.
3. Derive the horizontal asymptote of a rational function by dividing numerator and denominator by the highest power of the variable.
4. Explain the Intermediate Value Theorem and its role in the bisection method.
5. Discuss why continuity of a mathematical model does not necessarily imply smooth, stable, or physically accurate system behavior.

Analytical / Application Questions

1. Determine the value of k that makes f(x) = (x²−9)/(x−3) continuously extendable at x = 3, and state the extended value.
2. A controller uses u(r) = −10 for r < −10, u(r)=r for −10 ≤ r ≤ 10, and u(r)=10 for r > 10. Determine whether the function is continuous at both transition points and discuss one practical implication.
3. A sensor model is s(t)=t² cos(1/t) for t ≠ 0 and s(0)=0. Use the squeeze theorem to determine whether the model is continuous at t=0.

Multiple Choice Questions with Answers

Question 1

What is lim_x→2(x²−4)/(x−2)?

A. 0

B. 2

C. 4

D. Does not exist

Correct Answer: C. 4

Question 2

For a two-sided limit to exist at x=a, which condition is necessary?

A. The function must be defined at a

B. The left-hand and right-hand limits must be equal

C. The function must be differentiable at a

D. The function must have a vertical asymptote at a

Correct Answer: B. The left-hand and right-hand limits must be equal

Question 3

Which expression is an indeterminate form?

A. 5/2

B. 0/7

C. 0/0

D. 1+1

Correct Answer: C. 0/0

Question 4

What is lim_x→0sin(5x)/x when angles are measured in radians?

A. 0

B. 1/5

C. 5

D. Does not exist

Correct Answer: C. 5

Question 5

If lim_x→∞f(x)=3, which line is a horizontal asymptote?

A. x=3

B. y=3

C. y=0

D. x=0

Correct Answer: B. y=3

Question 6

For f(x)=1/(x−3), what happens as x→3⁺?

A. The function approaches 0

B. The function approaches 1

C. The function approaches +∞

D. The function approaches −∞

Correct Answer: C. The function approaches +∞

Question 7

Which condition is NOT required for continuity of f at x=a?

A. f(a) is defined

B. The limit at a exists

C. The limit equals f(a)

D. The derivative at a exists

Correct Answer: D. The derivative at a exists

Question 8

The squeeze theorem can be applied when a function is bounded between two functions that:

A. Have opposite infinite limits

B. Have the same limit

C. Are both discontinuous

D. Have equal derivatives only at one point

Correct Answer: B. Have the same limit

Question 9

For (4x²−x+7)/(2x²+3), what is the horizontal asymptote as x→∞?

A. y=0

B. y=1/2

C. y=2

D. y=4

Correct Answer: C. y=2

Question 10

What does the Intermediate Value Theorem guarantee for a continuous function whose endpoint values have opposite signs?

A. Exactly one root exists

B. At least one root exists between the endpoints

C. The derivative is zero between the endpoints

D. The function is monotonic between the endpoints

Correct Answer: B. At least one root exists between the endpoints