When finding the limit of a function at a certain point, the difference between finding the limit from the left and the limit from the right is based on the direction from which we approach the point on the function's domain.

1. Limit from the Left (Left-hand limit):

This is the limit of a function as the input approaches a specific value from values less than that point. It is written as: $\lim_{x \to c^-} f(x)$ This means that as $x$ approaches $c$ from values smaller than $c$, what value does $f(x)$ approach?

2. Limit from the Right (Right-hand limit):

This is the limit of a function as the input approaches the same value, but from values greater than that point. It is written as: $\lim_{x \to c^+} f(x)$ This means that as $x$ approaches $c$ from values larger than $c$, what value does $f(x)$ approach?

Key Point:

• One-sided limits: The left-hand and right-hand limits are referred to as "one-sided" limits. If both the left-hand and right-hand limits at a point exist and are equal, then the two-sided limit (just $\lim_{x \to c} f(x)$) exists and is equal to this common value.

• If $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$, the two-sided limit does not exist at $x = c$.

Example:

Consider the function:

1 & \text{if } x < 0 \\ 2 & \text{if } x \geq 0 \end{cases}$$ - Left-hand limit at $$x = 0$$: $$\lim_{x \to 0^-} f(x) = 1$$ - Right-hand limit at $$x = 0$$: $$\lim_{x \to 0^+} f(x) = 2$$ Since the left-hand and right-hand limits are not equal, $$\lim_{x \to 0} f(x)$$ does not exist. ### Do you want more details or have any further questions? ### 5 related questions: 1. What is a **two-sided limit** and how does it relate to one-sided limits? 2. Can a function be continuous if the one-sided limits are different? 3. How do you interpret limits graphically on a piecewise function? 4. What happens if a limit approaches infinity from one side but a finite value from the other? 5. How do discontinuities affect one-sided limits? ### Tip: Always check one-sided limits when dealing with piecewise or step functions to determine if a two-sided limit exists.