g. lim x sec x
h. lim
x 
x4
( x  h) 3  x 3
h 0
h
cos t  cos 2 t
t 0
t
i. lim
k.
lim 𝑒
𝑥→1
𝑥−1
5  x 1
2 x
j. lim
𝑥2 − 4
𝑙. lim 3
𝑥→−2 𝑥 + 8
𝜋𝑥
𝑠𝑖𝑛
2
3. Find the limits using any appropriate method.
 2 x  3, x  1
3 x
a. lim
b. lim f ( x)   2
x 1
x 3 x  3
x , x  1
𝑐. lim+ ln(𝑠𝑖𝑛𝑥)
𝑥→0
4. Find the x-values (if any) at which f is not continuous. Identify any discontinuities as removable or nonremovable. (Section 2.4)
ln( x  1), x  0
x2
a) f ( x)  2
b) f ( x)  
2
x  3 x  10
1  x , x  0
6. Find any vertical asymptotes for a) y 
x 1
x2 1
b) y 
x 1
x2 1
c) 𝑦 = csc(𝜋𝑥)
7. Use the Squeeze Theorem to find
1
lim 𝑥𝑐𝑜𝑠
𝑥→0
𝑥
8. Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c
guaranteed by the theorem.
𝑓(𝑥) = 𝑥 2 + 𝑥 − 2
[0,3]
f(c)=0
True or False
9. A rational function can have infinitely many x-values at which it is not continuous.
10. If lim 𝑓(𝑥) = 𝐿 then f(c)=L
𝑥→𝑐
11. Sketch the graph of a function such that the limit as x approaches 3 from the left = 2 and the limit
as x approaches 3 from the right = 0. Is the function continuous at x =2?