What is the limit as x approaches 0 of (sin(x^2)+1)^{1/(5x^2)} ?

The limit as x approaches 0 of (sin(x^2)+1)^{1/(5x^2)} is \sqrt[5]{e}

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limit as x approaches 0 of (sin(x^2)+1)^{1/(5x^2)}

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

Decimal Notation

1.22140 \dots

Solution steps

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

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{5 x ^{2}} l n (s i n (x^{2}) + 1)})

Apply the Limit Chain Rule:

\frac{d}{d x} (0.882 x^{0.842})

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

\frac{x + x cos ( x ) - 2 sin ( x )}{x ^{3}}

\frac{d}{d x} ((2 x - 5)^{4} (8 x^{2} - 5)^{- 3})

\frac{\partial}{\partial u} (\frac{uv}{v + 1})

What is the limit as x approaches 0 of (sin(x^2)+1)^{1/(5x^2)} ?
The limit as x approaches 0 of (sin(x^2)+1)^{1/(5x^2)} is \sqrt[5]{e}