# Analytical Approaches to Finding Limits: Factoring, Rationalizing, and Algebraic Techniques

Topics: Calculus

## Finding Limits Analytically

The limit of a function x approaches 3 as x approaches 3. We will call that x squared. If we
graph this, it looks something like this. And there are tick marks, but we will not worry about
them. The value 3 is over here and we have a point on the on curve approaching 9
Let f(x) = x2. Then f(3) = 9, so the limit of f(x) as x approaches 3 is also 9. This proves that f
is continuous at x = 3.
[Graph]
When we take the limit as x approaches 3 of this function fox, we get 3 squared is 9 plus 3
is 12. 4 times 3 is 12; but when we subtract 12 from both sides, we are left with 0.
In this case, we are unable to determine the limit of the function at this point. More
investigation is needed.
The first example will be solved by factoring. Another example is going to be solved by
rationalizing, or multiplying by a conjugate. Another example can be solved using FOIL or
distributing.
Algebra can be used to solve problems with an indeterminate form, in which you have a 0 in
the numerator and/or denominator.
lim_(x->3); x^2=3^2=9
f(x)=x^2
f(3)=9
f(x)=(x^2 -4x+3)/(x-3) f(x) Undef when x=3
lim_(x->3) (x^2 -4x+3)/(x-3) 0/0
Indeterminate form
– Factor
Algebra
– Rationalization / mult conjugate
-FOIL / distribute
– common denominator