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Limits

# The limit of a function

Let f be a function defined on an open interval containing a, except possibly at a itself.

Then $\lim_{x\to\text{a}}\text{f(x)}=L$ (read as the limit of f(x) as x approaches a is L) if for any $\varepsilon$>0, there exists a $\delta$>0

such that |f(x) - L|<$\varepsilon$ whenever |x-a|<$\delta$

Properties of limits

Given $\lim_{x\to\text{a}}\text{f(x)}=L$ and $\lim_{x\to\text{a}}\text{g(x)}=M$ and L, M, a, c, and n are real numbers, then:

1. $\lim_{x\to\text{a}}\text{c}=c$

2. $\lim_{x\to\text{a}}\text{cf(x)}=c\lim_{x\to\text{a}}\text{f(x)}=cL$

3. $\lim_{x\to\text{a}}\text{f(x)+g(x)}=\lim_{x\to\text{a}}\text{f(x)}+\lim_{x\to\text{a}}\text{g(x)}=L+M$

4. $\lim_{x\to\text{a}}\text{f(x)\;\cdot\;g(x)}=\lim_{x\to\text{a}}\text{f(x)}\;\cdot\;\lim_{x\to\text{a}}\text{g(x)}=L\;\cdot\;M$

5. $\lim_{x\to\text{a}}\frac{f(x)}{g(x)}=\frac{\lim_{x\to\text{a}}\text{f(x)}}{\lim_{x\to\text{a}}\text{g(x)}}=\frac{L}{M},\;M\;\neq\;0$

6. $\lim_{x\to\text{a}}\text{(f(x))^n}=(\lim_{x\to\text{a}}\text{f(x)})^n=L^n$