Limits and Continuity from a Graph – Step-by-Step Guidance

Q1. What is the left-hand limit of $f(x)$ as $x$ approaches $-1$?

Background

Topic: Limits from a Graph

This question tests your ability to interpret the graph of a function and determine the left-hand limit as $x$ approaches a specific value.

Key Terms and Formulas:

• Left-hand limit: The value that $f(x)$ approaches as $x$ comes from values less than $-1$.

• $\lim_{x \to -1^-} f(x)$: Limit as $x$ approaches $-1$ from the left.

Step-by-Step Guidance

1. Locate $x = -1$ on the graph. Identify the behavior of the function as $x$ approaches $-1$ from values less than $-1$ (moving rightward toward $-1$).

2. Observe the curve or points immediately to the left of $x = -1$. Look for the $y$-value that the function is approaching as $x$ gets closer to $-1$ from the left.

3. Check for any jumps, holes, or discontinuities at $x = -1$. The left-hand limit depends only on the values as you approach from the left, not the actual value at $x = -1$.

4. Identify the $y$-value the function is approaching and write it as $\lim_{x \to -1^-} f(x)$.

Try solving on your own before revealing the answer!

Final Answer: $-5$

As $x$ approaches $-1$ from the left, the graph approaches the value $-5$.

This is the $y$-value the function gets close to as you move toward $x = -1$ from the left side.