`lim_(x->0^(-)) (cothx)` Find the limit

Thursday, December 6, 2012

$\displaystyle\lim_{{{x}\to{{0}}^{{-}}}}{\left({\coth{{x}}}\right)}$ Find the limit

By definition, hyperbolic cotangent $\displaystyle{\coth{{\left({x}\right)}}}$ is equal to $\displaystyle\frac{{{\cosh{{\left({x}\right)}}}}}{{{\sinh{{\left({x}\right)}}}}},$ which is equal to $\displaystyle\frac{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}.$

When $\displaystyle{x}\to{{0}}^{{-}},$ both $\displaystyle{{e}}^{{x}}$ and $\displaystyle{{e}}^{{-{x}}}$ tend to $\displaystyle{1}.$ But while $\displaystyle{{e}}^{{x}}\to{{1}}^{{-}},$ $\displaystyle{{e}}^{{-{x}}}=\frac{{1}}{{{e}}^{{x}}}\to{1}+.$ Therefore the numerator tends to $\displaystyle{1}+{1}={2},$ and the denominator tends to $\displaystyle{{1}}^{{-}}-{{1}}^{+}={{0}}^{{-{.}}}$

And finally $\displaystyle\frac{{2}}{{{0}}^{{-{}}}}$ gives the limit of $\displaystyle-\infty.$ This is the answer.

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Thursday, December 6, 2012

$\displaystyle\lim_{{{x}\to{{0}}^{{-}}}}{\left({\coth{{x}}}\right)}$ Find the limit

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Thursday, December 6, 2012

Find the limit

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Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle\lim_{{{x}\to{{0}}^{{-}}}}{\left({\coth{{x}}}\right)}$ Find the limit

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?