Finding limits algebraically  direct substitution
Finding limits algebraically  direct substitution
Graphically finding the limit of a function is not always easy, as an alternative, we now shift our focus to finding the limit of a function algebraically. In this section, we will learn how to apply direct substitution to evaluate the limit of a function.
Lessons
Notes:
• if: a function $f$ is continuous at a number $a$
then: direct substitution can be applied: $\lim_{x \to a^} f(x) = \lim_{x \to a^+} f(x) =\lim_{x \to a} f(x)= f(a)$
• Polynomial functions are continuous everywhere, therefore “direct substitution” can ALWAYS be applied to evaluate limits at any number.

1.
No more finding limits “graphically”; Now, finding limits “algebraically”!

b)
When to apply Direct Substitution, and why Direct Substitution makes sense.
Exercise: $f(x)=\frac{1}{x2}$
i) Find the following limits from the graph of the function.
$\lim_{x \to 3} f(x)$
$\lim_{x \to 2.5} f(x)$
$\lim_{x \to 0} f(x)$
ii) Evaluate:
$f(3)$
$f(2.5)$
$f(0)$

2.
Evaluate the limit:

3.
Evaluate the onesided limit:

a)
$\lim_{x \to {1^  }} g(x)$
$\lim_{x \to {1^ + }} g(x)$
$\lim_{x \to {1}} g(x)$ 
b)
$\lim_{x \to {4^  }} g(x)$
$\lim_{x \to {4^ + }} g(x)$
$\lim_{x \to {4}} g(x)$
