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Use the definitions of the hyperbolic functions find each of the following limits.

(a) $ \displaystyle \lim_{x \to ^x} \tanh x $

(b) $ \displaystyle \lim_{x \to ^{-x}} \tanh x $

(c) $ \displaystyle \lim_{x \to ^x} \sinh x $

(d) $ \displaystyle \lim_{x \to ^{-x}} \sinh x $

(e) $ \displaystyle \lim_{x \to ^x} sech x $

(f) $ \displaystyle \lim_{x \to ^x} \coth x $

(g) $ \displaystyle \lim_{x \to 0^+} \coth x $ (h) $ \displaystyle \lim_{x \to 0^-} \coth x $

(i) $ \displaystyle \lim_{x \to ^{-x}} csch x $

(j) $ \displaystyle \lim_{x \to ^x} \frac {\sinh x}{e^x} $

a. $\frac{1-0}{1+0}=1$

b. $\frac{0-1}{0+1}=-1$

c. $\lim _{x \rightarrow \infty} \frac{e^{x}-e^{-x}}{2}=\infty$

d. $\lim _{x \rightarrow-\infty} \frac{e^{x}-e^{-x}}{2}=-\infty$

e. $\lim _{x \rightarrow \infty} \frac{2}{e^{x}+e^{-x}}=0$

f. $\frac{1+0}{1-0}=1$

g. $\lim _{x \rightarrow 0^{+}} \frac{\cosh x}{\sinh x}=\infty$

h. $\lim _{x \rightarrow 0^{-}} \frac{\cosh x}{\sinh x}=-\infty$

i. $\lim _{x \rightarrow-\infty} \frac{2}{e^{x}-e^{-x}}=0$

j. $\frac{1-0}{2}=\frac{1}{2}$

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