CALCULUS I
Solutions to Practice Problems
Limits
Paul Dawkins
Calculus I
Preface ............................................................................................................................................ 2
Limits.............................................................................................................................................. 2
Rates of Change and Tangent Lines......................................................................................................... 2
The Limit..................................................................................................................................................12
One-Sided Limits .....................................................................................................................................20
Limit Properties.......................................................................................................................................27
Computing Limits ....................................................................................................................................36
Infinite Limits ..........................................................................................................................................43
Limits At Infinity, Part I...........................................................................................................................56
Limits At Infinity, Part II .........................................................................................................................68
Continuity.................................................................................................................................................75
The Definition of the Limit......................................................................................................................90
© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
Calculus I
Preface
Here are the solutions to the practice problems for my Calculus I notes. Some solutions will have
more or less detail than other solutions. The level of detail in each solution will depend up on
several issues. If the section is a review section, this mostly applies to problems in the first
chapter, there will probably not be as much detail to the solutions given that the problems really
should be review. As the difficulty level of the problems increases less detail will go into the
basics of the solution under the assumption that if you’ve reached the level of working the harder
problems then you will probably already understand the basics fairly well and won’t need all the
explanation.
This document was written with presentation on the web in mind. On the web most solutions are
broken down into steps and many of the steps have hints. Each hint on the web is given as a
popup however in this document they are listed prior to each step. Also, on the web each step can
be viewed individually by clicking on links while in this document they are all showing. Also,
there are liable to be some formatting parts in this document intended for help in generating the
web pages that haven’t been removed here. These issues may make the solutions a little difficult
Limits
Rates of Change and Tangent Lines
1. For the function ( ) ( )
2
fx x = + 3 2 and the point P given by x = −3 answer each of the
following questions.
(a) For the points Q given by the following values of x compute (accurate to at least 8
decimal places) the slope, mPQ , of the secant line through points P and Q.
(i) -3.5 (ii) -3.1 (iii) -3.01 (iv) -3.001 (v) -3.0001
(vi) -2.5 (vii) -2.9 (viii) -2.99 (ix) -2.999 (x) -2.9999
(b) Use the information from (a) to estimate the slope of the tangent line to f x( ) at x = −3
and write down the equation of the tangent line.
© 2007 Paul Dawkins 2 http://tutorial.math.lamar.edu/terms.aspx
Calculus I
(a) For the points Q given by the following values of x compute (accurate to at least 8 decimal
places) the slope, mPQ , of the secant line through points P and Q.
(i) -3.5 (ii) -3.1 (iii) -3.01 (iv) -3.001 (v) -3.0001
(vi) -2.5 (vii) -2.9 (viii) -2.99 (ix) -2.999 (x) -2.9999
[Solution]
The first thing that we need to do is set up the formula for the slope of the secant lines. As
discussed in this section this is given by,
( ) ( )
( )
( )
2
33 2 3
3 3 PQ
fx f x
m
x x
−− + − = = −− +
Now, all we need to do is construct a table of the value of mPQ for the given values of x. All of
the values in the table below are accurate to 8 decimal places, but in this case the values
terminated prior to 8 decimal places and so the “trailing” zeros are not shown.
x mPQ x mPQ
-3.5 -7.5 -2.5 -4.5
-3.1 -6.3 -2.9 -5.7
-3.01 -6.03 -2.99 -5.97
-3.001 -6.003 -2.999 -5.997
-3.0001 -6.0003 -2.9999 -5.9997
(b) Use the information from (a) to estimate the slope of the tangent line to f x( ) at x = −3 and
write down the equation of the tangent line.
[Solution]
From the table of values above we can see that the slope of the secant lines appears to be moving
towards a value of -6 from both sides of x = −3 and so we can estimate that the slope of the
tangent line is : m = −6 .
The equation of the tangent line is then,
y f mx = − + − − = − + ⇒ =− − ( 3 3 3 6 3 ) ( ( )) ( x ) y x6 15
Here is a graph of the function and the tangent line.
© 2007 Paul Dawkins 3 http://tutorial.math.lamar.edu/terms.aspx
Calculus I
2. For the function gx x ( ) = + 4 8 and the point P given by x = 2 answer each of the
following questions.
(a) For the points Q