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Math 112 Solutions For Section 2.4, p. 142

1.
(a) 1. (c) -1. (f) $ \frac{\pi}{2}$. (g) 0. (h) 2. (k) 0. (l) 1. (m) 0.
3.
$ \sin x$ is bounded and $ \frac{1}{x}\to 0$ as $ x\to \infty$; use Theorem 34.
5.
(b) -1 (e) 0. (f) 1 (h) 1
8.
(b) $ y=1$ (c) none (d) $ y=\frac{\pi}{2}$ and $ y=-\frac{\pi}{2}$

9.
(b) $ \infty$. (c) DNE. (d) $ -\infty$. (g) $ \infty$. (h) $ -\infty$.

10.
(b) $ F\to \infty$ and $ G\to \infty$ as $ x\to \infty$, $ F\to
-\infty$ and $ G\to -1$ as $ x\to -\infty$.

11.
(c) $ x^{3}\to \infty$ as $ x\to \infty$ and $ x^{3}\to -\infty$ as $ x\to -\infty$. (d) $ x^{4}-x^{3}\to\infty$ as $ x\to\pm\infty$. (e) $ e^{x}\to\infty$ as $ x\to \infty$ and $ e^{x}\to 0$ as $ x\to -\infty$. (f) $ e^{-x^{2}}\to 0$ as $ x\to\pm\infty$. (g) $ \ln\vert x\vert\to\infty$ as $ x\to\pm\infty, \ln\vert x\vert\to-\infty$ as $ x\to 0$. (i) $ x^{2/3}\to \infty$ as $ x\to\pm\infty$.

12.
(a) $ x=1$. (b) $ x=2$. (c) $ x=2$ and $ x=4$. (d) $ x=0$.

13.
(c) vertical: $ x=1$; horizontal: $ y=0$.

14.
(b) $ \lim_{x\to c^{-}}f(x)=\infty$ if and only if, for any $ B>0$ there exists $ \delta > 0$ such that if $ 0<c-x<\delta$ then $ f(x)>B$.





Jason Grout 2003-01-31