Limits, Continuity and Differentiability (original) (raw)

Last Updated : 21 Feb, 2026

Limits, Continuity, and Differentiation are fundamental concepts in calculus. They are essential for analyzing and understanding functional behavior and are crucial for solving real-world problems in physics, engineering, and economics.

Limits

Limits are a fundamental concept in calculus that describes the behavior of a function as it approaches a certain point. Understanding limits is crucial for studying and understanding more complex ideas in calculus, such as continuity and differentiability. The limit of a function f(x) as x approaches a is the value that f(x) gets closer to as x approaches a.

Notation: lim⁡x→af(x) = L

Key Characteristics of Limits:

**Example of Limits:

Lim⁡x→2(3x+1) = 7

Continuity

Continuity of a function at a point means that the function is uninterrupted, or seamless, at that point. For a function to be continuous at a point, the limit of the function as it approaches that point must equal the function’s value at that point. A function f(x) is continuous at a point a if:

f(a) is defined

  1. Lim⁡x→af(x) exists
  2. Lim⁡x→af(x) = f(a)

Characteristics of Continuous Functions:

**Example of Continuity:

The function f(x) = x2 is continuous at all points.

Differentiability

Differentiability refers to the ability of a function to have a derivative at every point within its domain. A function is differentiable at a point if it has a defined slope at that point. A function f(x) is differentiable at a point a if its derivative exists at that point. The derivative represents the rate of change of the function.

**Notation: f′(a) = lim⁡h→0f(a+h) - f(a) / h

Properties of Differentiable Functions:

**Example of Differentiability:

f(x) = x2, f'x = 2x.

Interconnection Between Limits, Continuity, and Differentiability

Understanding the relationship between these concepts is pivotal:

Applications of Limits, Continuity, and Differentiability

Applications of engineering mathematics are essential across various engineering fields, enabling the solution of complex problems and the design of innovative systems.

  1. **Structural Engineering: Uses calculus to ensure the stability of structures by calculating stress and strain.
  2. **Electrical Engineering: Employed Fourier transforms and complex numbers for circuit analysis and design.
  3. **Mechanical Engineering: Applies differential equations to design and analyze machinery and thermodynamic systems.
  4. **Control Systems: Utilizes linear algebra and differential equations to develop controllers for dynamic systems such as robots and aircraft.
  5. **Fluid Mechanics: Leverages vector calculus to predict fluid behavior in applications like aerodynamics and pipeline flow.

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Solved Examples

**Example 1: Limit of a rational function, find lim(x→2) (x2 - 4) / (x - 2).

As x approaches 2, both numerator and denominator approach 0. Let's factor the numerator:
lim(x→2) (x2 - 4) / (x - 2) = lim(x→2) (x + 2)(x - 2) / (x - 2)
The (x - 2) cancels out:
= lim(x→2) (x + 2) = 2 + 2 = 4

**Example 2: Limit at infinity, find lim(x→∞) (3x2 + 2x - 1) / (x^2 + 5).

Divide both numerator and denominator by the highest power of x (x^2):
lim(x→∞) (3 + 2/x - 1/x2) / (1 + 5/x2)
As x approaches infinity, 1/x and 1/x2 approach 0:
= 3 / 1 = 3

**Example 3: One-sided limits, find the left-hand and right-hand limits of f(x) = |x| / x as x approaches 0.

Left-hand limit (x approaching 0 from negative side):
lim(x→0-) |x| / x = lim(x→0-) -x / x = -1
Right-hand limit (x approaching 0 from positive side):
lim(x→0+) |x| / x = lim(x→0+) x / x = 1
The left-hand and right-hand limits are not equal, so the limit does not exist.

**Example 4: Continuity at a point, determine if f(x) = { x^2 if x ≤ 2, 4x - 4 if x > 2 } is continuous at x = 2.

For continuity at x = 2, we need:
f(2) exists
lim(x→2) f(x) exists
f(2) = lim(x→2) f(x)
f(2) = 22= 4
Left-hand limit: lim(x→2-) x2 = 4
Right-hand limit: lim(x→2+) (4x - 4) = 4
f(2) = 4 = lim(x→2) f(x)
All conditions are satisfied, so f(x) is continuous at x = 2.

**Example 5: Differentiability, determine if f(x) = |x| is differentiable at x = 0.

For differentiability, the left-hand and right-hand derivatives must exist and be equal.
Left-hand derivative:
lim(h→0-) [f(0+h) - f(0)] / h = lim(h→0-) (|-h| - 0) / h = lim(h→0-) -h / h = -1
Right-hand derivative:
lim(h→0+) [f(0+h) - f(0)]/ h = lim(h→0+) (|h| - 0) / h = lim(h→0+) h / h = 1
The left-hand and right-hand derivatives are not equal, so f(x) is not differentiable at x = 0.

**Example 6: L'Hôpital's Rule, find lim(x→0) (sin x) / x.

This is a 0/0 indeterminate form, so we can apply L'Hôpital's Rule:
lim(x→0) (sin x) / x = lim(x→0) (d/dx sin x) / (d/dx x) = lim(x→0) cos x / 1 = 1

**Example 7: Intermediate Value Theorem, show that the equation x3 - x - 1 = 0 has at least one real root between 1 and 2.

Let f(x) = x3 - x - 1
f(1) = 13 - 1 - 1 = -1 (negative)
f(2) = 23 - 2 - 1 = 5 (positive)
Since f is continuous and changes sign between 1 and 2, by the Intermediate Value Theorem, there must be at least one point c between 1 and 2 where f(c) = 0.

Practice Problems

**Question 1. Evaluate the limit: lim(x→3) (x2 - 9) / (x - 3).

**Question 2. Find the limit, if it exists: lim(x→0) (sin(3x) / x).

**Question 3. Determine if the following function is continuous at x = 2:

**Question 4. Find the values of a and b that make the following function continuous everywhere:

**Question 5. Evaluate the limit using L'Hôpital's Rule: lim(x→∞) (ln(x) / x).

**Question 6. Determine if the function f(x) = |x - 1| is differentiable at x = 1.

**Question 7. Use the Intermediate Value Theorem to show that the equation x3 - 2x - 5 = 0 has at least one real root between 2 and 3.

**Question 8. Apply the Mean Value Theorem to the function f(x) = x3 on the interval [0, 2].

**Question 9. Find the limit: lim(x→0) (1 - cos(x)) / x2.

**Question 10. Prove that the function f(x) = 1/x is not uniformly continuous on the interval (0, 1).