Indeterminate Forms

Indeterminate Forms#

Definition#

Definition

If limx→af(x)=0 and limx→ag(x)=0, then

limx→af(x)g(x)

is called an indeterminate form of type 0/0.

How to Evaluate an Indeterminate Form of Type 0/0

To evaluate an indeterminate form, simplify the ratio f(x)/g(x) by factoring or rationalizing the expression and then canceling out common factors. Then try to evaluate the limit again using the simplified ratio.

Example 1#

An indeterminate form involving a rational function

Evaluate limx→5x−5x2−25.

Step 1:   Evaluate the limit of numerator and denominator.

We evaluate these limits by plugging in x=5:

limx→5x−5=5−5=0    and     limx→5x2−25=52−25=0

This means that the given limit is an indeterminate form of type 0/0, so we need to simplify the function before we can evaluate the limit.

Step 2:   Factor numerator and/or denominator and simplify.
x−5x2−25=x−5(x−5)(x+5)since A2−B2=(A−B)(A+B)=1x+5assuming x≠5

When computing the limit as x approaches 5, we are initially assuming that x is not equal to 5. This means that we can replace x−5x2−25 with 1x+5 when computing the limit, as shown in the next step.

Step 3:   Evaluate the limit using the simplified function.
limx→5x−5x2−25=limx→51x+5=15+5plug in x=5=110simplify

Example 2#

An indeterminate form involving a square root function

Evaluate limx→10x−6−2x−10.

Step 1:   Evaluate the limit of numerator and denominator.

Plug in x=10:

limx→10x−6−2=10−6−2=0    and     limx→10x−10=10−10=0

This means that the given limit is an indeterminate form of type 0/0, so we need to simplify the function before we can evaluate the limit.

Step 2:   Simplify the function.

We simplify the function by multiplying and dividing by x−6+2, which is the conjugate of x−6−2.

x−6−2x−10⋅x−6+2x−6+2=(x−6−2)(x−6+2)(x−10)(x−6+2)=x−6x−6+2x−6−2x−6−4(x−10)(x−6+2)FOIL=x−6+2x−6−2x−6−4(x−10)(x−6+2)simplify=x−6−4(x−10)(x−6+2)=x−10(x−10)(x−6+2)=1x−6+2if x≠10
Step 3:   Evaluate the limit using the simplified function.
limx→10x−6−2x−10=limx→101x−6+2=110−6+2plug in x=10=14+2simplify=14