Question

How do I solve Limit ((e^x)-(e^sin(x))/x-sin(x)) x->0
without using l'Hopital's rule?

Answer 1

Find the limit of \[\frac{ e ^{x}-e ^{\sin x} }{x -sinx } \] as x approaches 0?

Answer 2

if it is true, my suggestion is that you add 1 and subtract 1 in the numerator and separate them as \[\frac{ e ^{x}-1}{ x-sinx}+\frac{ 1-e ^{sinx} }{ x-sinx }\] and evaluate each of them as x approaches 0. But each of them is not going to give you a number. It's going to give you an expression which you can then combine the two limits again. Finally, the combination will give you 1 as the result.

Answer 3

and also use the hint : limit of \[\frac{ e ^{x} -1}{ x }\] as x->0 = 1