1. Evaluating limits
grade 12
Katleho phatoli 201125989

2. Sec. 1.3: Evaluating Limits
Analytically
 The limit of f(x) as x approaches c does not depend on
the value of f at c.
i.e. The limit of f(x) as x approaches c may not be f(c).
 Although, for those that are, we could have used
direct substitution to evaluate the limit.

3. Limits Using Direct
Substitution
If b and c are real numbers and n is a positive
integer, then
1.lim 2.lim 3.lim n n
x c x c x c
b b x c x c

4. More Limits Using Direct
Substitution
If p is a polynomial function, then
If r is a rational function r(x) = p(x)/q(x), then
lim ( ) ( )
x c
p x p c
( )
lim ( ) ( ) ( ) 0
( )x c
p c
r x r c q c
q c

5. More Limits Using Direct
Substitution
For radical functions, if n is positive, then the following
limit is valid for all c if n is odd, and all c > 0 if n is even.
lim n n
x c
x c

6. More Limits Using Direct
Substitution
For trigonometric functions, if c is in the domain of the
function, then
1. limsin sin 2. limcos cos
3. lim tan tan 4. limcot cot
5. limsec sec 6. limcsc csc
x c x c
x c x c
x c x c
x c x c
x c x c
x c x c

7. Properties of Limits (Rules)
Let b and c be real numbers, let n be a positive
integer, and let f and g be functions with the
following limits.
lim ( ) lim ( )
1.Scalar multiple: lim ( )
2.Sum or difference:lim ( ) ( )
3. Product: lim ( ) ( )
( )
4. Quotient: lim , 0
( )
5. Power: lim ( )
x c x c
x c
x c
x c
x c
n n
x c
f x L and g x K
b f x b L
f x g x L K
f x g x L K
f x L
K
g x K
f x L

8. More Properties of Limits
The limit of a composite function:
If f and g are functions such that
then
lim ( ) lim ( ) ( )
x c x L
g x L and f x f L
lim ( ( )) lim ( ) ( )
x c x c
f g x f g x f L

9. What if direct substitution
won’t work?
In a group 4 of research what happens if direct substation
doesn’t work is evaluating a limit.
The research is due after a week