UsefulLimits < MSC

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---+ Useful Limits

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...for working with sequences and series.
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---++ Exponentials

<latex> \lim_{h \rightarrow 0} (1+h)^{\frac{1}{h}} = e </latex>

<latex> \lim_{n \rightarrow \infty} (1+\frac{1}{n})^n = e </latex>

<latex> \lim_{n \rightarrow \infty} (1-\frac{1}{n})^n = e^{-1} </latex>

<latex> \lim_{n \rightarrow \infty} (1+\frac{x}{n})^n = e^x </latex>

---++ Powers and Roots

<latex> \lim_{n \rightarrow \infty} \frac{1}{n^p} =0 \mbox{ if $p>0$}</latex>

<latex> \lim_{n \rightarrow \infty} x^n \left \{ \begin{array}{l} \mbox{ $= 0$  if  $|x| <1$} \\ \mbox{diverges  if $ |x| >1 $} \end{array} \right.</latex>

<latex> \lim_{n \rightarrow \infty} \frac{x^n}{n!} = 0 \mbox{ for all $x$}</latex>

<latex> \lim_{n \rightarrow \infty} x^{\frac{1}{n}} = 1 \mbox{ for $x > 0$}</latex>

<latex> \lim_{n \rightarrow \infty} n^{\frac{1}{n}} = 1 </latex>

---++ Properties of Limits
---+++ The limit of a sum is the sum of the limits:
<latex> \lim_{n \rightarrow \infty} (a_n + b_n) = \lim_{n \rightarrow \infty} a_n + \lim_{n \rightarrow \infty} b_n </latex>

---+++ The limit of a product is the product of the limits:
<latex> \lim_{n \rightarrow \infty} (a_n \cdot b_n) = \lim_{n \rightarrow \infty} a_n \cdot \lim_{n \rightarrow \infty} b_n </latex>

---+++ The limit of a quotient is the quotient of the limits:
<latex> \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \frac{\lim_{n \rightarrow \infty} a_n}{\lim_{n \rightarrow \infty} b_n} </latex>

---++ Squeeze Theorem:
<latex>\begin{array}{c}\mbox{If $a_n \le b_n \le c_n$ and $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} c_n = L$} \\ \mbox{then $\lim_{n \rightarrow \infty} b_n = L$}\end{array}</latex>

---+++ AKA the "Flyswatter Theorem". Why?

<latex>\begin{array}{c}\mbox{Series $b_n$ buzzes between $a_n$ and $c_n$ } \\ \mbox{$a_n$ and $c_n$ swat the buzzing at $L$ as $n \rightarrow \infty$}\end{array}</latex> :-) 

---++ Ratios of Polynomials

The limiting behavior of the ratio of two polynomials depends on the degree of the polynomials. The polynomial of higher degree "wins". If their degrees are the same, then the limit is the ratio of the leading coefficients.

<latex> \lim_{x \rightarrow \infty} \frac{p_0 + p_1x +p_2x^2 + ... +p_nx^n}{q_0 + q_1x + q_2x^2 + ... + q_mx^m}</latex>

<latex>= \lim_{x \rightarrow \infty} \frac{P(x)}{Q(x)} </latex>

<latex> =  \left \{ \begin{array}{l} \mbox{diverges  if the degree of $P$ is larger ($m < n$).} \\ \mbox{$\frac{p_n}{q_n}$ if the degree of $P$ and $Q$ are the same ($m = n$).} \\ \mbox{ $0$  if  the degree of $Q$ is larger ($m > n$).} \end{array} \right.</latex>

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-- Main.DickFurnas - 16 Nov 2008