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

The limit as x approaches infinity of (2^x+x)^{1/x} is 2

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limit as x approaches infinity of (2^x+x)^{1/x}

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

2

Solution steps

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

Apply exponent rules

= x \to \infty lim (e^{\frac{1}{x} l n (2^{x} + x)})

Apply the Limit Chain Rule:

2

= 2

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

x \to 2 lim (2 x^{3} - 10 x)

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

x \to 2 lim (5^{2} + 3)

x \to 1 lim (x^{a} - 1)

What is the limit as x approaches infinity of (2^x+x)^{1/x} ?
The limit as x approaches infinity of (2^x+x)^{1/x} is 2