Section 2.2 — Limit of a Function

Determine the following limits.

  1. \( \displaystyle \lim_{x \to -2^+} \frac{e^x}{x+2} \)
  2. \( \displaystyle \lim_{x \to 3^-} \frac{x-5}{x-3} \)
  3. \( \displaystyle \lim_{x \to 1^+} \ln(x^2-1) \)
  4. Sketch a graph of a function \(f\) that satisfies all of the following conditions:
    1. \( f(-2) = 1 \)
    2. \( f(3) = 4 \)
    3. \( \displaystyle \lim_{x \to -2^- } f(x) = 1 \)
    4. \( \displaystyle \lim_{x \to -2^+ } f(x) = -1 \)
    5. \( \displaystyle \lim_{x \to 3} f(x) = 2 \)
  5. Sketch a graph of a function \(f\) that satisfies all of the following conditions:
    1. \( f(-1) = -3 \)
    2. \( f(2) = 4 \)
    3. \( \displaystyle \lim_{x \to 2^- } f(x) = 1 \)
    4. \( \displaystyle \lim_{x \to 2^+ } f(x) = 4 \)
    5. \( \displaystyle \lim_{x \to -1} f(x) = -2 \)
    6. \( \displaystyle \lim_{x \to 4^- } f(x) = -\infty \)
    7. \( \displaystyle \lim_{x \to 4^+ } f(x) = \infty \)
    8. \( \displaystyle \lim_{x \to \infty } f(x) = 1 \)
  6. The graph of a function \( f \) is given below. Find the following.
    Graph of y = f(x) for problem 6
    Given graph of \(y=f(x)\).
    (a) \( \displaystyle \lim_{x \to 2^- } f(x) = \)
    (b) \( \displaystyle \lim_{x \to 2^+ } f(x) = \)
    (c) \( \displaystyle \lim_{x \to 2} f(x) = \)
    (d) \( f(2) = \)
    (e) \( \displaystyle \lim_{x \to 4^- } f(x) = \)
    (f) \( \displaystyle \lim_{x \to 4^+ } f(x) = \)
    (g) \( f(4) = \)
    (i) \( \displaystyle \lim_{x \to 7 } f(x) = \)
    (j) \( f(7) = \)
    (k) \( \displaystyle \lim_{x \to 9^- } f(x) = \)
    (l) \( \displaystyle \lim_{x \to 9^+ } f(x) = \)
    (m) \( \displaystyle \lim_{x \to 0} f(x) = \)
    (n) \( f(0) = \)
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