The limit as x approaches to infinity where c is a constant

Question

I can't figure out the limit of

$$\lim\limits_{x \to \infty}1-\left (1-\left(\frac{c}{x}\right)^x \right)$$

as x approaches to infinity. Can someone help me out?

Answer 1

$$\lim_{x\to\infty} (c/x)^x = 0$$ So your limit becomes 0.

If you meant this instead: $$\lim_{x\to\infty} 1-(1-\frac{c}{x})^x$$ Then it is $$\lim_{x\to\infty} 1-((1-\frac{c}{x})^{\frac{x}{-c}})^{-c}$$ $$=1-e^{-c}$$

Answer 2

HINT

The limit indicated in the OP is well defined for $c\ge 0$, and it suffices to prove that

$$\left(\frac c x \right)^x\to 0$$

indeed for $x>2c \implies \frac c x<\frac12$ and

$$0\le \left(\frac c x \right)^x\le \left(\frac12 \right)^x \to 0$$

For the following

$$\lim_{x\to\infty} 1-\left(1-\frac{c}{x}\right)^x$$

we have

$$\left[\left(1-\frac{c}{x}\right)^\frac{x}{c}\right]^c\to e^{-c}$$