## 1-6: Calculating Limits Using the Limit Laws

Limit Laws Suppose that $\displaystyle c$ is a constant and the limits

$\displaystyle \lim_{x\rightarrow a}f(x)$ and $\displaystyle \lim_{x\rightarrow a}g(x)$

exist. then

1$\displaystyle \lim_{x\rightarrow a}\left [f(x)+g(x) \right ]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$

2$\displaystyle \lim_{x\rightarrow a}\left [f(x)-g(x) \right ]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}g(x)$

3.$\displaystyle \lim_{x\rightarrow a}\left [cf(x)\right ]=c\lim_{x\rightarrow a}f(x)$

4.$\displaystyle \lim_{x\rightarrow a}\left [f(x)g(x) \right ]=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$

5.$\displaystyle \lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a} g(x)}$ if $\displaystyle \lim_{x\rightarrow a}g(x)\neq 0$

These five laws can be stated verbally as follows:

Sum Law 1. The limit of a sum is the sum of the limits

Difference Law 2. The limit of a difference is the difference of the limits

Constant Multiple Law 3. The limit of a constant times a function is the constant times the limit of the function

Product Law 4. The limit of a product is the product of the limits

Quotient Law 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0

Power Law 6$\displaystyle \lim_{x\rightarrow a}\left [ f(x) \right ]^{n}=\left [ \lim_{x\rightarrow a}f(x) \right ]^{n}$ where $\displaystyle n$ is a positive integer

7$\displaystyle \lim_{x\rightarrow a}c=c$

8. $\displaystyle \lim_{x\rightarrow a}x=a$

9$\displaystyle \lim_{x\rightarrow a}x^{n}=a^{n}$ where $\displaystyle n$ is a positive integer

10$\displaystyle \lim_{x\rightarrow a}\sqrt[n]{x}=\sqrt[n]{a}$ where $\displaystyle n$ is a positive integer (If $\displaystyle n$ is even, we assume that $\displaystyle a> 0$)

Root Law 11$\displaystyle \lim_{x\rightarrow a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x\rightarrow a}f(x)}$ where $\displaystyle n$ is a positive integer

(If $\displaystyle n$ is even, we assume that $\displaystyle \lim_{x\rightarrow a}f(x)> a$)

Direct Substitution Property If $\displaystyle f$ is a polynomial or a rational function and $\displaystyle a$ is in the domain of $\displaystyle f$ then

$\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$

1.Theorem  $\displaystyle \lim_{x\rightarrow a}f(x)=L$ if and only if $\displaystyle \lim_{x\rightarrow a^{-}}f(x)=L=\lim_{x\rightarrow a^{+}}f(x)$

2. Theorem  If $\displaystyle f(x)\leqslant g(x)$ when $\displaystyle x$ is near $\displaystyle a$ (except possibly at $\displaystyle a$) and the limits of $\displaystyle f$ and $\displaystyle g$ both exist as $\displaystyle x$ approaches $\displaystyle a$ then

$\displaystyle \lim_{x\rightarrow a}f(x)\leqslant \lim_{x\rightarrow a}g(x)$

3. The Squeeze Theorem  If $\displaystyle f(x)\leqslant g(x)\leqslant h(x)$ when $\displaystyle x$ is near $\displaystyle a$ (except possibly at $\displaystyle a$) and
$\displaystyle \lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a} h(x)=L$
then $\displaystyle \lim_{x\rightarrow a}g(x)=L$

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