Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 5
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41Worksheet # 4: Limits: A Numerical and Graphical Approach, the
Limit Laws
1. Comprehension check:
(a) In words, describe what “ lim
f (x) = L” means.
x
a
(b) In words, what does “ lim
f (x) =
” mean?
x
a
(c) Suppose lim
f (x) = 2. Does f (1) = 2?
x
1
(d) Suppose f (1) = 2. Does lim
f (x) = 2?
x
1
2. Let p(t) denote the distance (in meters) to the right of the origin of a particle at time t minutes after
3
noon. If p(t) = p(t) = t
45t, give the average velocity on the intervals [2, 2.1] and [2, 2.01]. Use this
information to guess a value for the instantaneous velocity of particle at 12:02pm.
2
2
3. Consider the circle x
+ y
= 25 and verify that the point ( 3, 4) lies on the circle. Find the slope of
the secant line that passes through the points with x-coordinates
3 and
3.1. Find the slope of the
secant line that passes through the points with x-coordinates
3 and
2.99.
Use this information to guess the slope of the tangent line to the circle at ( 3, 4). Write the equation
of the tangent line and use a graphing calculator to verify that you have found the tangent line.
4. Compute the value of the following functions near the given x value. Use this information to guess
the value of the limit of the function (if it exists) as x approaches the given value.
sin(2x)
4x
9
3
(a) f (x) =
, x =
(c) f (x) =
, x = 0
2x 3
2
x
x
(b) f (x) =
, x = 0
(d) f (x) = sin(π/x), x = 0
x
2
x
if x
0
5. Let f (x) =
.
x
1 if 0 < x and x = 2
3
if x = 2
(a) Sketch the graph of f .
(b) Compute the following:
i. lim
f (x)
iii. lim
f (x)
vi. lim
f (x)
x
0
x
0
x
2
iv. f (0)
vii. lim
f (x)
x
2
ii. lim
f (x)
v. lim
f (x)
viii. f (2)
x
0
x
2
6. Compute the following limits or explain why they fail to exist:
x + 2
x + 2
(a)
lim
(c) lim
x + 3
x + 3
x
3
x
3
x + 2
1
(b)
lim
(d) lim
x + 3
3
x
x
3
x
0
7. Given lim
f (x) = 5 and lim
g(x) = 2, use limit laws to compute the following limits or explain why
x
2
x
2
we cannot find the limit. Note when working through a limit problem that your answers should be a
chain of true equalities. Make sure to keep the lim
operator until the very last step.
x
a
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