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Numerade Educator

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Problem 23 Medium Difficulty

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} $

Answer

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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Video Transcript

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