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:
(i) \( f(-2) = 1 \), (ii) \(f(3)=4 \), (iii) \( \displaystyle \lim_{x \to -2^- } f(x) =1 \), (iv) \( \displaystyle \lim_{x \to -2^+ } f(x) = -1 \), (v) \( \displaystyle \lim_{x \to 3} f(x) = 2 \)
5.Sketch a graph of a function \(f\) that satisfies all of the following conditions.
(i) \( f(-1) = -3 \), (ii) \(f(2)= 4 \), (iii) \( \displaystyle \lim_{x \to 2^- } f(x) =1 \), (iv) \( \displaystyle \lim_{x \to 2^+ } f(x) = 4 \), (v) \( \displaystyle \lim_{x \to -1} f(x) = -2 \) \hfill\\ (vi) \( \displaystyle \lim_{x \to 4^- } f(x) = -\infty \), (vii) \( \displaystyle \lim_{x \to 4^+ } f(x) = \infty \), (viii) \( \displaystyle \lim_{x \to \infty } f(x) = 1\).
6. The graph of a function \( f \) is given below. Find the following.
(a) \( \displaystyle \lim_{x \to 2^- } f(x) = \qquad \qquad\) (b) \( \displaystyle \lim_{x \to 2^+ } f(x) = \qquad \qquad\) (c) \( \displaystyle \lim_{x \to 2} f(x) = \qquad \qquad\) (d) \( f(2)= \qquad \qquad\)
(e) \( \displaystyle \lim_{x \to 4^- } f(x) = \qquad \qquad\) (f) \( \displaystyle \lim_{x \to 4^+ } f(x) = \qquad \qquad\) (g) \( f(4)= \)
(i) \( \displaystyle \lim_{x \to 7 } f(x) = \qquad \qquad\) (j) \( f(7)= \qquad \qquad \)
(k) \( \displaystyle \lim_{x \to 9^- } f(x) = \qquad \qquad \) (l) \( \displaystyle \lim_{x \to 9^+} f(x) = \qquad \qquad \)
(m) \( \displaystyle \lim_{x \to 0} f(x) = \qquad \qquad\) (n) \( \displaystyle f(0) =\qquad \qquad \)
Solution.pdf
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:
(i) \( f(-2) = 1 \), (ii) \(f(3)=4 \), (iii) \( \displaystyle \lim_{x \to -2^- } f(x) =1 \), (iv) \( \displaystyle \lim_{x \to -2^+ } f(x) = -1 \), (v) \( \displaystyle \lim_{x \to 3} f(x) = 2 \)
5.Sketch a graph of a function \(f\) that satisfies all of the following conditions.
(i) \( f(-1) = -3 \), (ii) \(f(2)= 4 \), (iii) \( \displaystyle \lim_{x \to 2^- } f(x) =1 \), (iv) \( \displaystyle \lim_{x \to 2^+ } f(x) = 4 \), (v) \( \displaystyle \lim_{x \to -1} f(x) = -2 \) \hfill\\ (vi) \( \displaystyle \lim_{x \to 4^- } f(x) = -\infty \), (vii) \( \displaystyle \lim_{x \to 4^+ } f(x) = \infty \), (viii) \( \displaystyle \lim_{x \to \infty } f(x) = 1\).
6. The graph of a function \( f \) is given below. Find the following.
(a) \( \displaystyle \lim_{x \to 2^- } f(x) = \qquad \qquad\) (b) \( \displaystyle \lim_{x \to 2^+ } f(x) = \qquad \qquad\) (c) \( \displaystyle \lim_{x \to 2} f(x) = \qquad \qquad\) (d) \( f(2)= \qquad \qquad\)
(e) \( \displaystyle \lim_{x \to 4^- } f(x) = \qquad \qquad\) (f) \( \displaystyle \lim_{x \to 4^+ } f(x) = \qquad \qquad\) (g) \( f(4)= \)
(i) \( \displaystyle \lim_{x \to 7 } f(x) = \qquad \qquad\) (j) \( f(7)= \qquad \qquad \)
(k) \( \displaystyle \lim_{x \to 9^- } f(x) = \qquad \qquad \) (l) \( \displaystyle \lim_{x \to 9^+} f(x) = \qquad \qquad \)
(m) \( \displaystyle \lim_{x \to 0} f(x) = \qquad \qquad\) (n) \( \displaystyle f(0) =\qquad \qquad \)
Solution.pdf