Limits and Continuity

Limits and Continuity

Limits and continuity are fundamental concepts in calculus. They help us understand how functions behave as they approach certain values. This understanding is crucial for solving problems in mathematics, physics, and engineering. By mastering these concepts, you will be better prepared to tackle more complex topics in differentiation and calculus.

Understanding Limits

A limit describes the value that a function approaches as the input approaches a certain point. For example, consider the function f(x) = x². If we want to find the limit of f(x) as x approaches 2, we can evaluate f(x) for values close to 2.

Evaluating Limits

To evaluate the limit of f(x) = x² as x approaches 2, we can calculate:

x = 1.9, f(1.9) = 3.61
x = 1.99, f(1.99) = 3.9601
x = 1.999, f(1.999) = 3.996001
x = 2.1, f(2.1) = 4.41
x = 2.01, f(2.01) = 4.0401
x = 2.001, f(2.001) = 4.004001

As x approaches 2 from both sides, the values of f(x) approach 4. We can write:

lim_x→2 f(x) = 4.

Definition of a Limit

The formal definition of a limit is as follows:

If f is defined for all x near a (except possibly at a), and if f(x) gets arbitrarily close to L as x approaches a, we write:

lim_x→a f(x) = L.

One-Sided Limits

One-sided limits consider the behaviour of a function as x approaches a value from one side only. There are two types:

• Right-sided limit: lim_x→a⁺ f(x) = L means that x approaches a from the right.
• Left-sided limit: lim_x→a⁻ f(x) = L means that x approaches a from the left.

Example of One-Sided Limits

Consider the function:

f(x) = { -2x + 4, if x ≤ 1
  { √(x - 1), if x > 1

To find the limits at x = 1:

1. lim_x→1⁻ f(x) = -2(1) + 4 = 2
2. lim_x→1⁺ f(x) = √(1 - 1) = 0

Since the left-sided limit (2) does not equal the right-sided limit (0), we conclude:

lim_x→1 f(x) does not exist.

Continuity

A function is continuous at a point a if:

• f(a) is defined.
• lim_x→a f(x) exists.
• lim_x→a f(x) = f(a).

Example of Continuity

For the function f(x) = x² at x = 2:

• f(2) = 4 (defined).
• lim_x→2 f(x) = 4 (exists).
• lim_x→2 f(x) = f(2).

Thus, f(x) is continuous at x = 2.

Limits at Infinity

Limits can also be evaluated as x approaches infinity (∞) or negative infinity (-∞). For example:

f(x) = 5/x²

To find lim_x→-∞ f(x):

lim_x→-∞ (5/x²) = 5/(-∞)² = 0.

Example of Limits at Infinity

For the function:

f(x) = (x² - 1)/(x² + 1)

To evaluate lim_x→∞ f(x):

Divide numerator and denominator by x²:

limx→∞ (1 - 1/x²)/(1 + 1/x²) = 1.

Summary

• Limits describe the behaviour of functions as they approach specific values.
• One-sided limits consider the approach from one side only.
• Continuity requires that a function is defined, the limit exists, and they are equal.
• Limits can also be evaluated at infinity.

Check your understanding

1. What is the definition of a limit?
2. How do you determine if a limit exists at a point?
3. What is the difference between a left-sided limit and a right-sided limit?
4. Give an example of a function that is not continuous at a point and explain why.