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

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

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

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

1

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

1

= 1

x \to - 1 lim (∣ x - 1 ∣ (x + 1))

x \to 5 - lim (\frac{x ^{2} + 7}{x ^{2} - 25})

x \to 0 lim ((e^{x} - x)^{\frac{3}{x}})

x \to - \frac{π}{2} lim (- 7 x - \frac{π}{2})

x \to 1 lim (\frac{x - 1}{x ^{2} - 8 x + 7})

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