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Find the limit at the following points by looking at the graph below of f(x) . If the limit does not exist, the DNE

From the image you provided, I can observe the graph of the function \( f(x) \), and there are four limit questions asked based on the graph. Let's solve them by analyzing the behavior of the graph at the given points:

### (a) \( \lim_{x \to 4} f(x) \)
- From the graph, as \( x \to 4 \), the function approaches the value of \( y = 5 \). There is no discontinuity or jump at this point. 
  \[
  \boxed{5}
  \]

### (b) \( \lim_{x \to 2^-} f(x) \)
- For \( x \to 2 \) from the left (\( 2^- \)), the graph appears to approach \( y = 0 \). So, the left-hand limit is:
  \[
  \boxed{0}
  \]

### (c) \( \lim_{x \to 2^+} f(x) \)
- For \( x \to 2 \) from the right (\( 2^+ \)), the graph shows a jump, and the function value jumps to \( y = 5 \). So, the right-hand limit is:
  \[
  \boxed{5}
  \]

### (d) \( \lim_{x \to 2} f(x) \)
- Since the left-hand limit and the right-hand limit as \( x \to 2 \) are not equal (left-hand limit is 0 and right-hand limit is 5), the two-sided limit does not exist (DNE).
  \[
  \boxed{\text{DNE}}
  \]

### Summary of the Limits:
- \( \lim_{x \to 4} f(x) = 5 \)
- \( \lim_{x \to 2^-} f(x) = 0 \)
- \( \lim_{x \to 2^+} f(x) = 5 \)
- \( \lim_{x \to 2} f(x) = \text{DNE} \)

Would you like further details or explanations on any part? Here are 5 related questions to explore:

1. What is the definition of a one-sided limit?
2. How does continuity at a point relate to the existence of a two-sided limit?
3. Can a function have different one-sided limits and still be continuous at that point?
4. What is the significance of a jump discontinuity for limits?
5. How do we formally prove that a limit does not exist?

**Tip:** Always check both one-sided limits before concluding whether a two-sided limit exists!

Find the limit at the following points by looking at the graph of f(x). If the limit does not exist, the DNE.

Learn how to evaluate limits from a graph. This tutorial covers left-hand and right-hand limits, discontinuities, and determining whether two-sided limits exist. Detailed step-by-step solutions for finding limits at x=4 and x=2.

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