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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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