What is the limit as x approaches 0 of (e^{sin(x)}-e^{tan(x)})/(sin(x)-tan(x)) ?

The limit as x approaches 0 of (e^{sin(x)}-e^{tan(x)})/(sin(x)-tan(x)) is 1

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limit as x approaches 0 of (e^{sin(x)}-e^{tan(x)})/(sin(x)-tan(x))

x \to 0 lim (\frac{e ^{s i n (x)} - e ^{t a n (x)}}{sin ( x ) - tan ( x )})

1

Solution steps

x \to 0 lim (\frac{e ^{s i n (x)} - e ^{t a n (x)}}{sin ( x ) - tan ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{e ^{s i n (x)} cos ( x ) - e ^{t a n (x)} sec ^{2} ( x )}{cos ( x ) - sec ^{2} ( x )})

= x \to 0 lim (\frac{e ^{s i n (x)} cos ( x ) - e ^{t a n (x)} sec ^{2} ( x )}{cos ( x ) - sec ^{2} ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{e ^{s i n (x)} cos ^{2} ( x ) - e ^{s i n (x)} sin ( x ) - e ^{t a n (x)} sec ^{4} ( x ) - 2 e ^{t a n (x)} sec ^{2} ( x ) tan ( x )}{- sin ( x ) - 2 sec ^{2} ( x ) tan ( x )})

= x \to 0 lim (\frac{e ^{s i n (x)} cos ^{2} ( x ) - e ^{s i n (x)} sin ( x ) - e ^{t a n (x)} sec ^{4} ( x ) - 2 e ^{t a n (x)} sec ^{2} ( x ) tan ( x )}{- sin ( x ) - 2 sec ^{2} ( x ) tan ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{e ^{s i n (x)} cos ^{3} ( x ) - e ^{s i n (x)} sin ( 2 x ) - e ^{s i n (x)} cos ( x ) sin ( x ) - e ^{s i n (x)} cos ( x ) - e ^{t a n (x)} sec ^{6} ( x ) - 6 e ^{t a n (x)} sec ^{4} ( x ) tan ( x ) + 4 e ^{t a n (x)} sec ^{2} ( x ) - 6 e ^{t a n (x)} sec ^{4} ( x )}{- cos ( x ) + 4 sec ^{2} ( x ) - 6 sec ^{4} ( x )})

= x \to 0 lim (\frac{e ^{s i n (x)} cos ^{3} ( x ) - e ^{s i n (x)} sin ( 2 x ) - e ^{s i n (x)} cos ( x ) sin ( x ) - e ^{s i n (x)} cos ( x ) - e ^{t a n (x)} sec ^{6} ( x ) - 6 e ^{t a n (x)} sec ^{4} ( x ) tan ( x ) + 4 e ^{t a n (x)} sec ^{2} ( x ) - 6 e ^{t a n (x)} sec ^{4} ( x )}{- cos ( x ) + 4 sec ^{2} ( x ) - 6 sec ^{4} ( x )})

Plug in the value

x = 0

= \frac{e ^{s i n (0)} cos ^{3} ( 0 ) - e ^{s i n (0)} sin ( 2 \cdot 0 ) - e ^{s i n (0)} cos ( 0 ) sin ( 0 ) - e ^{s i n (0)} cos ( 0 ) - e ^{t a n (0)} sec ^{6} ( 0 ) - 6 e ^{t a n (0)} sec ^{4} ( 0 ) tan ( 0 ) + 4 e ^{t a n (0)} sec ^{2} ( 0 ) - 6 e ^{t a n (0)} sec ^{4} ( 0 )}{- cos ( 0 ) + 4 sec ^{2} ( 0 ) - 6 sec ^{4} ( 0 )}

Simplify

\frac{e ^{s i n (0)} cos ^{3} ( 0 ) - e ^{s i n (0)} sin ( 2 \cdot 0 ) - e ^{s i n (0)} cos ( 0 ) sin ( 0 ) - e ^{s i n (0)} cos ( 0 ) - e ^{t a n (0)} sec ^{6} ( 0 ) - 6 e ^{t a n (0)} sec ^{4} ( 0 ) tan ( 0 ) + 4 e ^{t a n (0)} sec ^{2} ( 0 ) - 6 e ^{t a n (0)} sec ^{4} ( 0 )}{- cos ( 0 ) + 4 sec ^{2} ( 0 ) - 6 sec ^{4} ( 0 )} : 1

= 1

x \to 0 lim (\frac{1}{x} + \frac{1}{t ^{2} + 1})

x \to 0 lim (\frac{( 4 sin ( \frac{π}{2} + x ) - 4 sin ( \frac{π}{2} ) )}{x})

x \to - 1 + lim (\frac{- 1}{x + 1})

x \to 2 lim ((x^{2} + 3) (x - 4))

x \to 1 - lim (\frac{- 3 ( x ^{2} - 1 )}{2 x ^{2} + 2 x - 4})

What is the limit as x approaches 0 of (e^{sin(x)}-e^{tan(x)})/(sin(x)-tan(x)) ?
The limit as x approaches 0 of (e^{sin(x)}-e^{tan(x)})/(sin(x)-tan(x)) is 1