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

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

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

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

1

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

1

= 1

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

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

x \to \infty lim (1 - x^{3})

x \to 0 lim (x + \frac{4}{x ^{2}})

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

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