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

The limit as x approaches 0 of (1+x^2)^{1/(1-cos(x))} is e^2

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limit as x approaches 0 of (1+x^2)^{1/(1-cos(x))}

x \to 0 lim ((1 + x^{2})^{\frac{1}{1 - c o s ( x )}})

e^{2}

Decimal

7.38905 \dots

Solution steps

x \to 0 lim ((1 + x^{2})^{\frac{1}{1 - c o s ( x )}})

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{1 - c o s ( x )} l n (1 + x^{2})})

Apply the Limit Chain Rule:

e^{2}

= e^{2}

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

x \to 10 lim (\frac{10 x}{2 x + 5})

x \to \frac{π}{4} lim (tan (2 x) tan (\frac{π}{4} - x))

x \to \infty lim (\frac{π}{2})

x \to - 2 lim (x - 3)

What is the limit as x approaches 0 of (1+x^2)^{1/(1-cos(x))} ?
The limit as x approaches 0 of (1+x^2)^{1/(1-cos(x))} is e^2