One-Sided Limits with Solved Example

One-sided limits in calculus refer to the value a function approaches as the input gets closer to a particular point from either the left or the right. When we analyze the function from the left side, it is called the left-hand limit, denoted as lim⁡_x→c−f(x), while from the right side, it's called the right-hand limit and is written as lim⁡_x→c+.

In this article, we will explore the concept of one-sided limits, work through examples, and provide practice problems to reinforce your understanding.

What is a One-Sided Limit?

A one-sided limit refers to the value a function approaches as the input approaches a particular value from only one side—either from the left (left-hand limit) or from the right (right-hand limit).

Importance of One-Sided Limits

• Discontinuities: One-sided limits help in analyzing functions that have discontinuities (like jumps or breaks). If the left-hand and right-hand limits at a point are not equal, the overall limit at that point does not exist.
• Piecewise Functions: One-sided limits are particularly useful for functions that have different definitions in different intervals, like piecewise functions.

Types of One-Sided Limit

There are two types of one-sided limit i.e.,

• Left-Hand Limit (LHL)
• Right-Hand Limit (RHL)

Let's dicuss these in detail.

Left-Hand Limit

The left-hand limit of a function f(x) as x approaches aaa is denoted as:

\lim_{x \to a^-} f(x)

This means that x approaches aaa from the left (values smaller than a).

Right-Hand Limit

The right-hand limit of a function f(x) as x approaches aaa is denoted as:

\lim_{x \to a^+} f(x)

This means that x approaches aaa from the right (values larger than a).

Note: If both one-sided limits exist and are equal at a point, then the two-sided limit at that point exists and is equal to the common value.

\lim_{x \to a} f(x) \quad \text{exists if and only if} \quad \lim_{x \to a^-} f(x) = \lim_{x \to a^+}

Let's consider an example for better understanding.

Consider the function: f(x) = \begin{cases} 2x + 1 & \text{if } x < 3 \\ x^2 - 4 & \text{if } x \geq 3 \end{cases}

To find the one-sided limits at x=3x = 3x=3:

• Left-hand limit: lim⁡_x→3−f(x) = 2(3) + 1 = 7
• Right-hand limit: lim_⁡x→3+f(x)=3² − 4 = 5

Since the left-hand limit (7) and right-hand limit (5) are not equal, the limit does not exist at x = 3. However, the one-sided limits still exist separately.

Read More about Limits.

Solved Examples on One-Sided Limits

Example 1: For f(x) = 3x - 1, find the left-hand limit as x approaches 2.

Solution:

We need to evaluate \lim_{x \to 2^-} f(x).

f(x) = 3x - 1

Substituting x = 2 directly into the function:

f(2) = 3(2) - 1 = 6 - 1 = 5

Since the function is linear and continuous, the left-hand limit is:

\lim_{x \to 2^-} f(x) = 5

Example 2: For f(x) = \frac{1}{x}, find the right-hand limit as x approaches 0.

Solution:

We need to find \lim_{x \to 0^+} \frac{1}{x}

For values of x approaching 0 from the right (positive values), \frac{1}{x} grows infinitely large. Therefore:

\lim_{x \to 0^+} \frac{1}{x} = +\infty

Example 3: For f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ 3x - 1 & \text{if } x \geq 1 \end{cases}, find the left-hand and right-hand limits as x approaches 1.

Solution:

• For the left-hand limit:

\lim_{x \to 1^-} f(x) = 1 + 2 = 3

• For the right-hand limit:

\lim_{x \to 1^+} f(x) = 3(1) - 1 = 2

Since the left-hand limit (3) and the right-hand limit (2) are different, the two-sided limit does not exist at x = 1.

Example 4: For f(x) = \begin{cases} 5 - x & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}, find the right-hand limit as x approaches 0.

Solution:

For the right-hand limit:

\lim_{x \to 0^+} f(x) = 0^2 = 0

Example 5: For f(x) = \frac{1}{x - 2}, find the left-hand and right-hand limits as x approaches 2.

Solution:

• For the left-hand limit:

\lim_{x \to 2^-} \frac{1}{x - 2} = -\infty

• For the right-hand limit:

\lim_{x \to 2^+} \frac{1}{x - 2} = + \infty

Since the limits from both sides are infinite but with opposite signs, the two-sided limit does not exist at x = 2.

Practice Questions

Q1: f(x) = x² - 4x + 3, find \lim_{x \to 1^-}.

Q2: f(x) = \frac{2}{x}, find \lim_{x \to 0^+} f(x).

Q3: f(x) = 3x - 1, find \lim_{x \to 3^+} f(x).

Q4: f(x) = \frac{1}{x - 1}​, find \lim_{x \to 1^-} f(x).

Q5: f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ x^2 - 4 & \text{if } x \geq 2 \end{cases}​, find \lim_{x \to 2^+} f(x).

Q6: f(x)=x³, find \lim_{x \to -1^-} f(x).

Q7: f(x) = \frac{1}{x + 3}​, find \lim_{x \to -3^+}.

Q8: f(x) = \frac{x^2}{x - 2}​, find \lim_{x \to 2^-} f(x).

Answer Key

1. f(1) = 0
2. +\infty
3. 8
4. -\infty
5. 0
6. −1
7. +\infty
8. 4

Conclusion

In conclusion, one-sided limits are an important concept in calculus that help us understand how a function behaves as it approaches a specific point from either the left or the right. They provide deeper insight, especially for functions that are not continuous or have jumps, gaps, or other types of discontinuities.

Read More,

• Calculus in Maths
• Differential Calculus
• Formal Definition of Limit
• Limit at Infinity