IXL - Find limits involving factorization and rationalization (Calculus practice)

Evaluate the limit by expanding the numerator.

lim

→0

(

–

)

–

Simplify any fractions and radicals in your answer.

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question

Evaluate the limit by expanding the denominator.

lim

→0

–

(

–

)

–

Simplify any fractions and radicals in your answer.

key idea

When evaluating a limit of the type

lim

→

(

)

(

)

you cannot use the limit law for division if

lim

→

(

)

lim

→

(

)

and

lim

→

(

)

you can try to rewrite

(

)

(

)

by first canceling an expression and then applying valid limit laws.

solution

First, see if the limit law for division can be used to evaluate the limit:

lim

→0

–

(

–

)

–

Find the limit of the denominator.

lim

→0

[

(

–

)

–

]

[

(

)

–

]

–

Since the limit of the denominator is 0, the limit law for division cannot be used.

Now, find the limit of the numerator.

lim

→0

–

(

)

Since the limits of the numerator and denominator are both 0, try to find expressions to cancel.

The denominator is a complicated expression and can be simplified. Start by expanding the denominator.

lim

→0

–

(

–

)

–

lim

→0

–

lim

→0

–

lim

→0

–

(

–

)

Cancel

lim

→0

–

Cancel

lim

→0

–

You can cancel

because as

→0

is never equal to 0. So,

Now, try to evaluate the limit.

lim

→0

–

(

)

Looking for more? Try this:

$Confetti$

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Teacher tools

Work it out

Not feeling ready yet? These can help:

X.4Simplify rational expressions

B.5Find limits of polynomials and rational functions

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