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For the function f graphed below, find the following limits:



1. \displaystyle \lim_{x \to -1^{-}}f(x) = 
help (limits)
2. \displaystyle \lim_{x \to -1^{+}}f(x) = 

3. \displaystyle \lim_{x \to 1} f(x) = 

4. \displaystyle \lim_{x \to \infty}f(x) = 

5. \displaystyle \lim_{x \to -\infty}f(x) = 



Let's go through the limits one by one, based on the graph provided.

1. **\(\lim_{x \to -1^{-}} f(x)\)**:
   - This represents the left-hand limit as \(x\) approaches \(-1\) from the left. From the graph, as \(x\) approaches \(-1\) from the left, the function is oscillating but remains bounded around \(0\). Thus:
     \[
     \lim_{x \to -1^{-}} f(x) = 0
     \]

2. **\(\lim_{x \to -1^{+}} f(x)\)**:
   - This represents the right-hand limit as \(x\) approaches \(-1\) from the right. From the graph, as \(x\) approaches \(-1\) from the right, the function diverges towards \(-\infty\). Therefore:
     \[
     \lim_{x \to -1^{+}} f(x) = -\infty
     \]

3. **\(\lim_{x \to 1} f(x)\)**:
   - This is the two-sided limit as \(x\) approaches \(1\). As \(x\) approaches \(1\), the function seems to diverge towards \(\infty\), implying that there is a vertical asymptote at \(x = 1\). Thus:
     \[
     \lim_{x \to 1} f(x) = \infty
     \]

4. **\(\lim_{x \to \infty} f(x)\)**:
   - As \(x\) approaches \(\infty\), the function settles towards a horizontal asymptote. From the graph, the function appears to approach \(0\) as \(x\) increases. Hence:
     \[
     \lim_{x \to \infty} f(x) = 0
     \]

5. **\(\lim_{x \to -\infty} f(x)\)**:
   - As \(x\) approaches \(-\infty\), the function oscillates with a decaying amplitude, and it seems to approach \(0\). Therefore:
     \[
     \lim_{x \to -\infty} f(x) = 0
     \]

### Summary:
- \(\lim_{x \to -1^{-}} f(x) = 0\)
- \(\lim_{x \to -1^{+}} f(x) = -\infty\)
- \(\lim_{x \to 1} f(x) = \infty\)
- \(\lim_{x \to \infty} f(x) = 0\)
- \(\lim_{x \to -\infty} f(x) = 0\)

Would you like further details or explanations on any of these limits?

Here are five related questions:
1. What is the difference between a left-hand limit and a right-hand limit?
2. How can we identify a vertical asymptote from a function's graph?
3. What conditions lead to an infinite limit as \(x\) approaches a certain value?
4. What is the significance of a horizontal asymptote in a function?
5. Can a function have multiple vertical asymptotes? Why or why not?

**Tip:** Always check the behavior of the function from both sides of a point when evaluating limits, especially near vertical asymptotes!

For the function f graphed below, find the following limits:

1. lim(x → -1^-) f(x)
2. lim(x → -1^+) f(x)
3. lim(x → 1) f(x)
4. lim(x → ∞) f(x)
5. lim(x → -∞) f(x)

Solve limits for a function involving vertical asymptotes and oscillatory behavior. This problem covers key calculus concepts like limits, vertical asymptotes, and horizontal asymptotes, with a focus on understanding the behavior as x approaches critical points. Suitable for high school students in Grades 10-12.

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