Try These Funky Limits!
lim (3x
1.
2
x →0
8 + x3
lim
2
x→ −2 x + 2 x
x4 −1
lim
3
x →1 1 − x
5.
lim9 − 5( x − 2)
x 2 − 6x + 7
8. lim 2
x →1 x − 8 x + 9
4.
7.
− 2 x − 7)
x2 − 5
2. lim
x →1 2 − x
x →2
10.
13.
x3 − 8
lim
2
x → 2 x − 9 x + 14
x+6−3
x−3
lim
x →3
16.
19.
11.
x 2 − 12 x + 36
14. lim
2 x − 12
x →6
17.
lim
20.
lim
x→4
x−4
x−4
lim 5
x
23.
lim
x →1
33.
lim
x →2
lim
29.
x − 3x − 2
5 − x − 2x
1 − 7 − 3x
2x + 5 − x + 7
2 2 x − 2 −2 x
35. lim 3 x
− 2 −3 x
x →0 2
1 − 4x
x − 2x
1
x−2
2 x 3 − 5x 2 + 3 x − 2
lim
2x − 4
x →2
x 3 − 2 x 2 + 23
26. lim
23 x 4 − 7
x →
x→ − 
31.
1
4
x→2+
42 x 3 − 2 x + 5
25. lim
6x3 − 7
x →
28.
lim
x→
x →0
22.
6.
2 x − 2−x
lim
x
−x
x→0 2 + 2
x 3 − 2x − 5
9. lim
2x3 − 7
x→2
12.
32.
3x
lim
x
x→+  4
lim
x →0
x +1 − 2x +1
3x + 4 − 2 x + 4
1
1
−
34. lim 3 + x 3
x
x →0
1
x3 − 2
36. lim
x −8
x →8
x−4
lim 2 −
x→4
h →0
( x + a) 2
lim
2
2
x→a x + a
2x
18 x − h
lim h − 6 x
x2 − 1
3. lim
x → −1 1 + x
x
81 − x 4
15. lim
x→ − 3 x + 3
−1
18.
lim x
21.
lim 2 x
24.
lim
x →0−
3
1
x →0−
−1
x→ − 
5x
27.
2 − x8
lim
7
x → 3 x − 11
30.
5x
lim
x
x → 4
Limits Help Sheet
Methods for finding limits:
• Using your brain - if the limit is obvious by looking at it, then just write the answer!
• Substitution - if you can substitute the value into the expression without encountering an indeterminate
form like 0 0 or   then go ahead and do it. Don’t waste time with the other methods!
• Factor - if you do have an indeterminate form, factor first, then divide out the troublesome term.
Note: you may have to factor expressions that aren’t as pretty as previous grades (like square roots etc..)
• Divide top & bottom by highest power of x - this only works if you are approaching +  or − 
(Use your horizontal asymptote strategies from Advanced Functions for this)
• Rationalize numerator, denominator or both - always choose the easier one first!
• Ingenuity - sometimes the best solution is to go with a gut feeling. Don’t be afraid to try something new!
Reminders:
• Make sure you know why you are doing something. If you don’t know, why bother?
• If you wish to verify your answer, you can try a value on your calculator, but this is only a guess/check!
• Most of the time, the algebraic limit is the same from the left and from the right, BUT NOT ALWAYS!
If you have an asymptote, a discontinuity (in the graph), or a domain restriction (like a square root or a
log), make sure you check both limits!
0
something
= 0 and
• Remember that
= DNE ( +  or −  ). Don’t make life difficult!
something
0
• If you can, picture the graph to help you figure out a limit.