What is the limit as x approaches infinity of \sqrt[x]{4^x+9^x} ?

The limit as x approaches infinity of \sqrt[x]{4^x+9^x} is 9

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limit as x approaches infinity of \sqrt[x]{4^x+9^x}

9

Solution steps

Apply exponent rules

= x \to \infty lim (e^{\frac{1}{x} l n (4^{x} + 9^{x})})

Simplify

e^{\frac{1}{x} l n (4^{x} + 9^{x})} : (4^{x} + 9^{x})^{\frac{1}{x}}

= x \to \infty lim ((4^{x} + 9^{x})^{\frac{1}{x}})

Apply exponent rules

= x \to \infty lim (e^{\frac{1}{x} l n (4^{x} + 9^{x})})

Apply the Limit Chain Rule:

9

= 9

\frac{d}{d x} (\frac{cos ( x )}{2})

- 2 x^{2}

\int 9 x^{2} - 8 x d x

\int \frac{3}{1 + 9 x ^{2}} d x

\int \frac{y ^{2} - 4}{y} d y

What is the limit as x approaches infinity of \sqrt[x]{4^x+9^x} ?
The limit as x approaches infinity of \sqrt[x]{4^x+9^x} is 9