# Tutor profile: Griffin J.

## Questions

### Subject: Set Theory

Let $f:X\to Y$ and let $B\subset Y$ and let $C\subset B$. Prove that $f^{-1}(C) \subset f^{-1}(B)$

We need to show that every point in $f^{-1}(C)$ is also in $f^{-1}(B)$. So take $x$ to be any point in $f^{-1}(C)$ Remember that for any $Y' \subset Y$, $f^{-1}(Y') = \{a | f(a) \in Y'\} Thus we have that $f(x) \in C$. Since $C\subset B$ we have that $f(x) \in B$. This implies that $x \in f^{-1}(B)$ and we are done.

### Subject: LaTeX

How do you line up multiple equations along their equal signs

Use the align environment and before each equal sign you have to put a & for example: \begin{align*} a &= b \\ c &= d \end{align*}

### Subject: Calculus

Prove that $$ \lim_{n\to \infty} (1 + \frac{1}{n})^n = e $$

Let $(1+\frac{1}{n})^n = y$ Then $$ \lim_{n\to\infty} \ln(y) = \lim_{n\to \infty} ln((1+\frac{1}{n})^n) = \lim_{n\to \infty} n\ln(1+\frac{1}{n}) $$ This last expression can be re-written as: \lim_{n\to \infty} \frac{\ln(1+\frac{1}{n})}{1\n}. Applying L'Hopital's Rule (since we would get the indeterminate from 0/0 if we were to evaluate this limit) we get $$ \lim_{n\to \infty} \frac{\ln(1+\frac{1}{n})}{1\n} = \lim_{n\to \infty} \frac{\frac{-n^{-2}}{1 + 1/n}}{-n^{-2}} $$ After canceling the $-n^{-2}$'s, this last expression gives us: $$ \lim_{n\to \infty}\frac{1}{1+1/n} = 1 $$ Thus we have that $$ \lim_{n\to\infty} \ln(y) = 1. $$ Therefore \lim_{n\to\infty}e^{\ln(y)} = e^1 = e

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