Limit
Use to study the output of a function as the input(s) approach something (a number or infinity)
Of the form
Read as:
has the limit as approaches is the limit as approaches of of is the limit of of as approaches
+
The limit must be a number
When the limit is equal to
Sidedness
By default,
For left-side (
Superscript the limit with
[!example]+ Left-side limit
has the left-side limit as approaches :
[!example]+ Right-side limit
has the left-side limit as approaches :
+ Existence of a limit theorem
Otherwise, we say the limit D.N.E. (does not exist)
The limit is approached but never reached
Properties & theorems
, where is any of addition, subtraction, multiplication or division (when = division, ) , where or and
Direct substitution property
If
Evaluation
For a generic limit
- If evaluating a piecewise function at boundary of two pieces, the left and right limits need to be considered;
- Substitute
into using the direct substitution property. If the result is an indeterminate form, follow the steps below.
Indeterminate forms
Multiple ways to solve
For a limit
where
first rationalize the fraction.
Because it is always possible to factorize a polynomial equal to 0, we factorize
- to
and cancel , or - with long division.
Now that it no longer contains the factor that makes it 0, we can solve the limit as usual.
For a limit
there are three possible answers:
when when when
99% of the time, forcefully factorize the dominant term.
Otherwise, rationalize.
[!example]-
When, limit = .
We factorize:Since
, .
- Factorize
- Combine
- Rationalize
[!example]-
if, limit = .
Combining the two fractions gives. Substituting
into gives .
Limit at infinity
- Forcefully factorize dominant term
- Rationalize
- Combine