# Limit laws

### Limit laws

Basic concepts: Function notation,

#### Lessons

Here are some properties of limits:

1) $\lim_{x \to a} x = a$
2) $\lim_{x \to a} c = c$
3) $\lim_{x \to a} [cf(x)] = c\lim_{x \to a}f(x)$
4) $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a}f(x) \pm \lim_{x \to a}g(x)$
5) $\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a}f(x) \lim_{x \to a}g(x)$
6) $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}$, only if $\lim_{x \to a}g(x) \neq0$
7) $\lim_{x \to a} [f(x)]^n=[\lim_{x \to a}f(x)]^n$

Where c is a constant, $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist.

Here is a fact that may be useful to you.
If $P(x)$ is a polynomial, then
$\lim_{x \to a} P(x)=P(a)$
• 1.
Evaluating Limits of Functions
Evaluate the following limits using the property of limits:
a)
$\lim_{x \to 2} x^2+4x+3$

b)
$\lim_{x \to 2} 3(x^2+4x+3)^2$

c)
$\lim_{x \to 1} \frac{2-3x+4x^2}{2+x^4}$

d)
$\lim_{x \to 0} 4(3)^x$

e)
$\lim_{x \to \frac{\pi}{2}} 3(\sin x)^4$

• 2.
Evaluating Limits with specific limits given
Given that $\lim_{x \to 5} f(x)=-3$, $\lim_{x \to 5} g(x)=5$, $\lim_{x \to 5} h(x)=2$, use the limit properties to compute the following limits:
a)
$\lim_{x \to 5} [5f(x)-2g(x)]$

b)
$\lim_{x \to 5} [g(x)f(x)+3h(x)]$

c)
$\lim_{x \to 5} \frac{2g(x)}{h(x)}$

d)
$\lim_{x \to 5} \frac{5[f(x)]^3}{g(x)}$