Prove that $f(x) = (1 + x)^{\frac{1}{x}}$ is continuous

Question

Prove that $f(x) = (1 + x)^{\frac{1}{x}}$ is continuous on the region $(-1, \infty) \subset \mathbb{R}.$

Attempt at a solution:

We need $|(1 + x)^{\frac{1}{x}} - (1 + y)^{\frac{1}{y}}| < \epsilon$ whenever $|x-y| < \delta.$ It seems like the proper course of action would be to take $$|x-y| < \epsilon^{xy}$$ $$|x-y|^{1/xy} < \epsilon$$ $$|(1+x)-(1+y)|^{1/xy} < \epsilon$$ And then show $$|(1 + x)^{\frac{1}{x}} - (1 + y)^{\frac{1}{y}}| < |(1+x)-(1+y)|^{1/xy}$$ Which is just proving that

$$|a^c - b^d| < |a-b|^{cd}$$ Without loss of generality we assume $a^c > b^d$ so we just need to show

$$a^c - b^d < |a-b|^{cd}$$ It seems like the binomial theorem is in order, but I'm not quite sure how to apply it here.

P.S. We know that $$\lim_{x\to 0} f(x) =e.$$ So just define $f(0) = \lim_{x\to 0} f(x).$

Answer 1

For $x \neq 0$ $$f(x)=e^{{ln(1+x)}\over x}$$ is continuous since it is a composition of continuous functions.

For $x=0$ just define $f(0) = \lim_{x\to 0} f(x)=e$.

Answer 2

$f(x)=e^{{ln(1+x)}\over x}$ it is a composition of continuous functions so it is continuous. $x \rightarrow ln(x+1)$ is continuous, $g(x)={{ln(x+1)}\over x}$ is continous since it is the quotient of two continuous function, so $e\circ g=f$ is continuous.