Understanding Left Hand Limit (LHL) and Right Hand Limit (RHL)

In calculus, the concepts of Left Hand Limit (LHL) and Right Hand Limit (RHL) are crucial for understanding the behavior of functions as they approach a certain point from different directions. These concepts are fundamental in determining the existence of limits and continuity of functions.

Definitions

The Left Hand Limit of a function $f(x)$ as $x$ approaches a point $c$ from the left is denoted by $f(c^-)$ or $\lim_{x \to c^-} f(x)$. It represents the value that the function approaches as the input approaches $c$ from values less than $c$.

The Right Hand Limit of a function $f(x)$ as $x$ approaches a point $c$ from the right is denoted by $f(c^+)$ or $\lim_{x \to c^+} f(x)$. It represents the value that the function approaches as the input approaches $c$ from values greater than $c$.

Formal Definitions

• LHL: $\lim_{x \to c^-} f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $0 < c - x < \delta$ implies $|f(x) - L| < \epsilon$.

• RHL: $\lim_{x \to c^+} f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $0 < x - c < \delta$ implies $|f(x) - L| < \epsilon$.

Table of Differences and Important Points

Aspect	Left Hand Limit (LHL)	Right Hand Limit (RHL)
Direction of Approach	Approaches the point $c$ from the left (i.e., from values less than $c$)	Approaches the point $c$ from the right (i.e., from values greater than $c$)
Notation	$f(c^-)$ or $\lim_{x \to c^-} f(x)$	$f(c^+)$ or $\lim_{x \to c^+} f(x)$
Definition	Value that $f(x)$ approaches as $x$ approaches $c$ from the left	Value that $f(x)$ approaches as $x$ approaches $c$ from the right
Existence	LHL exists if the function approaches a specific value as $x$ approaches $c$ from the left	RHL exists if the function approaches a specific value as $x$ approaches $c$ from the right
Continuity	If LHL exists and equals $f(c)$, it is necessary (but not sufficient) for continuity at $c$	If RHL exists and equals $f(c)$, it is necessary (but not sufficient) for continuity at $c$
Limit	The limit $\lim_{x \to c} f(x)$ exists if and only if LHL and RHL both exist and are equal	The limit $\lim_{x \to c} f(x)$ exists if and only if LHL and RHL both exist and are equal

Examples

Example 1: Step Function

Consider the step function:

$$ f(x) = \begin{cases} 1 & \text{if } x < 2 \ 3 & \text{if } x \geq 2 \end{cases} $$

Calculate the LHL and RHL at $x = 2$.

Solution:

• LHL at $x = 2$: $\lim_{x \to 2^-} f(x) = 1$ because as $x$ approaches 2 from the left, $f(x)$ remains at 1.
• RHL at $x = 2$: $\lim_{x \to 2^+} f(x) = 3$ because as $x$ approaches 2 from the right, $f(x)$ remains at 3.

Since the LHL and RHL are not equal, the limit $\lim_{x \to 2} f(x)$ does not exist, and the function is not continuous at $x = 2$.

Example 2: Piecewise Function

Consider the piecewise function:

$$ f(x) = \begin{cases} x^2 & \text{if } x < 1 \ 2x - 1 & \text{if } x \geq 1 \end{cases} $$

Calculate the LHL and RHL at $x = 1$.

Solution:

• LHL at $x = 1$: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1$.
• RHL at $x = 1$: $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x - 1) = 1$.

Since the LHL and RHL are equal, the limit $\lim_{x \to 1} f(x)$ exists and is equal to 1. However, for continuity at $x = 1$, we also need $f(1) = 1$. Since $f(1) = 2(1) - 1 = 1$, the function is continuous at $x = 1$.

Conclusion

Understanding LHL and RHL is essential for analyzing the behavior of functions near a point and determining the existence of limits and continuity. When both LHL and RHL exist and are equal, the limit of the function at that point exists. If the limit equals the function's value at that point, the function is continuous there.