What is the limit as x approaches 0 of (e^{x^2}-cos(x))/(sin^2(x)+3sin^5(x)) ?

The limit as x approaches 0 of (e^{x^2}-cos(x))/(sin^2(x)+3sin^5(x)) is 3/2

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limit as x approaches 0 of (e^{x^2}-cos(x))/(sin^2(x)+3sin^5(x))

x \to 0 lim (\frac{e ^{x^{2}} - cos ( x )}{sin ^{2} ( x ) + 3 sin ^{5} ( x )})

\frac{3}{2}

Decimal

1.5

Solution steps

x \to 0 lim (\frac{e ^{x^{2}} - cos ( x )}{sin ^{2} ( x ) + 3 sin ^{5} ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 e ^{x^{2}} x + sin ( x )}{sin ( 2 x ) + 15 sin ^{4} ( x ) cos ( x )})

= x \to 0 lim (\frac{2 e ^{x^{2}} x + sin ( x )}{sin ( 2 x ) + 15 sin ^{4} ( x ) cos ( x )})

Apply L'Hopital's Rule

: x \to 0 lim \frac{2 ( 2 e ^{x^{2}} x ^{2} + e ^{x^{2}} ) + cos ( x )}{cos ( 2 x ) \cdot 2 + 15 ( 4 sin ^{3} ( x ) cos ^{2} ( x ) - sin ^{5} ( x ) )}

= x \to 0 lim \frac{2 ( 2 e ^{x^{2}} x ^{2} + e ^{x^{2}} ) + cos ( x )}{cos ( 2 x ) \cdot 2 + 15 ( 4 sin ^{3} ( x ) cos ^{2} ( x ) - sin ^{5} ( x ) )}

Plug in the value

x = 0

= \frac{2 ( 2 e ^{0^{2}} \cdot 0 ^{2} + e ^{0^{2}} ) + cos ( 0 )}{cos ( 2 \cdot 0 ) \cdot 2 + 15 ( 4 sin ^{3} ( 0 ) cos ^{2} ( 0 ) - sin ^{5} ( 0 ) )}

Simplify

\frac{2 ( 2 e ^{0^{2}} \cdot 0 ^{2} + e ^{0^{2}} ) + cos ( 0 )}{cos ( 2 \cdot 0 ) \cdot 2 + 15 ( 4 sin ^{3} ( 0 ) cos ^{2} ( 0 ) - sin ^{5} ( 0 ) )} : \frac{3}{2}

= \frac{3}{2}

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

x \to - 1 + lim (2 x + 1)

x \to 0 + lim ((1 + 5 x)^{\frac{2}{x}})

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

x \to 1 lim (\frac{3 x}{x - 1})

What is the limit as x approaches 0 of (e^{x^2}-cos(x))/(sin^2(x)+3sin^5(x)) ?
The limit as x approaches 0 of (e^{x^2}-cos(x))/(sin^2(x)+3sin^5(x)) is 3/2