# 2.3 | Evaluating Limits Analytically

If $$c$$ is a real number, then
$\lim_{x \rightarrow a} c = c$
$\lim_{x \rightarrow a} x = a$
Limit Laws:

• Sum/Difference Law:
$\lim_{x \rightarrow a} \left( f(x) \pm g(x) \right) = \lim_{x \rightarrow a} f(x) \pm \lim_{x \rightarrow a} g(x)$
• Product Law:
$\lim_{x \rightarrow a} \left( f(x)g(x) \right) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x)$
• Constant Multiple Law:
$\lim_{x \rightarrow a} \left( cf(x) \right) = c\lim_{x \rightarrow a} f(x),$
for any $$c$$.
• Quotient Law:
$\lim_{x \rightarrow a} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \rightarrow a} f(x)}{ \lim_{x \rightarrow a} g(x)},$
provided that $$\lim_{x \rightarrow a} g(x) \not = 0$$.
• Root Law:
$\lim_{x \rightarrow a} \sqrt[n]{f(x)}= \sqrt[n]{\lim_{x \rightarrow a} f(x)},$
provided that $$n$$ is a positive integer, and $$\lim_{x \rightarrow a} f(x) >0$$ if $$n$$ is even.
If $$n$$ is a positive integer then
$\lim_{x \rightarrow a} \left(f(x)\right)^n = \left(\lim_{x \rightarrow a} f(x)\right)^n.$
$\lim_{x \rightarrow a} x^n = a^n$
If $$p(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots +a_0$$ is a polynomial function, then
$\lim_{x \rightarrow a} p(x) = p(a)$
If $$f$$ is a rational function defined by $$f(x) = P(x)/Q(x)$$, where $$P(x)$$ and $$Q(x)$$ are polynomial functions and $$Q(a) \not = 0$$, then
$\lim_{x \rightarrow a} f(x) = f(a) = \frac{P(a)}{Q(a)}$
Let $$a$$ be a number in the domain of the given trigonometric function. Then

• $$\displaystyle \lim_{x \rightarrow a} \sin x = \sin a$$
• $$\displaystyle \lim_{x \rightarrow a} \cos x = \cos a$$
• $$\displaystyle \lim_{x \rightarrow a} \tan x = \tan a$$
• $$\displaystyle \lim_{x \rightarrow a} \csc x = \csc a$$
• $$\displaystyle \lim_{x \rightarrow a} \sec x = \sec a$$
• $$\displaystyle \lim_{x \rightarrow a} \cot x = \cot a$$
The Squeeze Theorem: If $$h(x) \leq f(x) \leq g(x)$$ for all $$x$$ in an open interval containing $$c$$, except possibly at $$c$$, and
$\lim_{x \rightarrow c} h(x) = L = \lim_{x \rightarrow c} g(x)$
Then
$\lim_{x \rightarrow c} f(x) = L$
Suppose that $$f(x) \leq g(x)$$ for all $$x$$ in an open interval containing $$a$$, except possibly at $$a$$, and
$\lim_{x \rightarrow a}f(x) = L \text{ and } \lim_{x \rightarrow a} g(x) = M$
Then
$L \leq M$
$\lim_{\theta \rightarrow 0} \frac{\sin u}{u} = 1$
$\lim_{u \rightarrow 0} \frac{\cos u -1}{u} = 0$
$\lim_{u \rightarrow 0} (1+u)^{1/u} = e$