How do you solve #\lim _ { x \rightarrow 0} \frac { \sinh ( x ) - x } { x ^ { 3} }#?

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Answer 1 · 2017-11-06T15:28:44

#1/6#

#\lim _ { x \rightarrow 0} \frac { \sinh ( x ) - x } { x ^ { 3} }#

This limit has #0/0# uncertainity. After using L'Hospital rule,

#\lim _ { x \rightarrow 0} \frac { \cosh ( x ) - 1 } { 3x ^ { 2} }#

It has #0/0# uncertainity again. After using L'Hospital rule,

#\lim _ { x \rightarrow 0} \frac { \sinh ( x )} { 6x }#

It has #0/0# uncertainity again. After using L'Hospital rule,

#\lim _ { x \rightarrow 0} \frac { \cosh ( x )} { 6 }=1/6#