Chapter 2: Limits and Continuity
Magnet Game!
Find a value for a so that the function
is continuous.
 x  1, x  3
f ( x)  
2
ax
,
x

3

2
Evaluate the limit, if possible.
x  7 x  10
lim
x 5
x 5
2
Find the average rate of change of the
function over the interval [10,12].
f ( x)  4 x  1
Is the function continuous at x = 3?
If not, what type of discontinuity
exists?
Find the slope of the curve at x = 5.
f ( x)  x  4
Evaluate the limit, if possible.
tan  2 x 
lim
x 0
x
Find the slope of the curve at x =-3.
f ( x)  x
2
Evaluate the limit, if possible.
1
lim
x0 x
Find the slope of the curve at x = 1.
f ( x)  x  4
Evaluate the limit, if possible.
3x  2 x  1
lim
5
x  4 x  100
4
Is the function continuous at x = 1?
If not, what type of discontinuity
exists?
What is the
Three Step Test
to prove continuity at a
point?
Give a formula for the extended
function that is continuous at the
indicated point.
x 1
f  x  2 , x  1
x 1
3
Evaluate the limit, if possible.
x4
lim
x 4
x 2
Write the equation of the line normal
to the curve at x =3.
1
f ( x) 
x2
Evaluate the limit, if possible.
lim e
x 0
x
Write the equation of the line tangent
to the curve at x =3.
1
f ( x) 
x2
Evaluate the limit, if possible.
3a  x 

lim
x 0
2
x
 9a
2
Explain why the given function is
continuous.
g ( x) 
1
x 1
Evaluate the limit, if possible.
1
lim x  sin  
x 
x
Evaluate the limit, if possible.
1
1

x

4
4
lim
x 0
x
Evaluate the limit, if possible.
x 4 x
2
lim e
x 
4 x 100
5
Evaluate the limit, if possible.
1
lim
x 1 1  x
Evaluate the limit, if possible.
1
lim
x  x
Evaluate the limit, if possible.
sin  3x 
lim
x 0
x
The function is not continuous at x=2.
Which step in the Test for Continuity
fails?
Find the slope of the curve at x = 4.
f ( x)  x  4