# Questions tagged [exponential-function]

For question involving exponential functions and questions on exponential growth or decay.

530 questions
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### Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$ [duplicate]

Based on the definition of $e: = \lim_{x\to\infty} \left(1+\frac1x \right)^x$, how can we show that $$\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}?$$ So far I've tried ...
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### how does one solve $\lim_{n\to \infty} (\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$

I cant use lhopital's rule here is my attempt: Limit of $(\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$ =$e^{\lim_{n\to \infty} n\times \ln(\frac{1 -2-n +\frac{1}{n^{2}}}{1 -\frac{4}{n} +\frac{2}{n^{2}}})}$ I ...
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### Solution of trigonometric equation $0 = x\cos(x)+2$

I'm having a hard time finding the solution of the following equation: $$0 = x\cos(x)+2$$ This is part of showing that $f(x):= x^2e^{\sin(x)}$ has 3 extremas in $[-\frac{\pi}{2},\frac{\pi}{2}]$.
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### Series proof for $e^x$.

Problem: Prove $$\sum_{n=0}^\infty \frac{1}{n!}x^n=e^x$$ I am a bit confused on how I should start this proof. Any pointers on how I should start would help.
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### Solving an exponential inequality problem

How do I prove the following inequality : $$\Bigg(\frac{2}{\alpha^2} \, \big( e^{\alpha x} - e^{\alpha y} \big) \, + \, e^{\alpha y} (y^2 - x^2) \; \Bigg) > 0$$ given, $x, y > 0$ ? Can ...
### Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B$ onto $\mathbb{C} \setminus \{0\}$.
Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B$ onto $\mathbb{C} \setminus \{0\}$ where $B$ is a bounded subset of $\mathbb{C}$. I have no idea how to even attempt this question so I would ...