Chapter 3
Limits and Rates of Change
L
imits are the foundation of calculus. Being able to work with limits and to understand
them conceptually is crucial, because key ideas and definitions in calculus make use
of limits. This chapter examines a variety of limit problems and makes the intuitive idea
of continuity formal by using limits. Many later problems also involve the use of limits, so
although limits may go away for a while during your calculus studies, they’ll return!
The Problems You’ll Work On
In this chapter, you encounter a variety of problems involving limits:
✓ Using graphs to find limits
✓ Finding left-hand and right-hand limits
✓ Determining infinite limits and limits at infinity
✓ Practicing many algebraic techniques to evaluate limits of the form 0/0
✓ Determining where a function is continuous
What to Watch Out For
You can use a variety of techniques to evaluate limits, and you want to be familiar with them
all! Remember the following tips:
✓ When substituting in the limiting value, a value of zero in the denominator of a fraction
doesn’t automatically mean that the limit does not exist! For example, if the function
has a removable discontinuity, the limit still exists!
✓ Be careful with signs, as you may have to include a negative when evaluating limits at
infinity involving radicals (especially when the variable approaches negative infinity).
It’s easy to make a limit positive when it should have been negative!
✓ Know and understand the definition of continuity, which says the following: A function
f(x) is continuous at a if lim ( ) ( )
x a
f x
f a
→
=
.
Limits and Rates of Change
L
imits are the foundation of calculus. Being able to work with limits and to understand
them conceptually is crucial, because key ideas and definitions in calculus make use
of limits. This chapter examines a variety of limit problems and makes the intuitive idea
of continuity formal by using limits. Many later problems also involve the use of limits, so
although limits may go away for a while during your calculus studies, they’ll return!
The Problems You’ll Work On
In this chapter, you encounter a variety of problems involving limits:
✓ Using graphs to find limits
✓ Finding left-hand and right-hand limits
✓ Determining infinite limits and limits at infinity
✓ Practicing many algebraic techniques to evaluate limits of the form 0/0
✓ Determining where a function is continuous
What to Watch Out For
You can use a variety of techniques to evaluate limits, and you want to be familiar with them
all! Remember the following tips:
✓ When substituting in the limiting value, a value of zero in the denominator of a fraction
doesn’t automatically mean that the limit does not exist! For example, if the function
has a removable discontinuity, the limit still exists!
✓ Be careful with signs, as you may have to include a negative when evaluating limits at
infinity involving radicals (especially when the variable approaches negative infinity).
It’s easy to make a limit positive when it should have been negative!
✓ Know and understand the definition of continuity, which says the following: A function
f(x) is continuous at a if lim ( ) ( )
x a
f x
f a
→
=
.