10th Edition

Ron Larson + 1 other

ISBN: 9781285057095

Textbook Problem

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary.

lim x → ∞ e x x 4

To determine

To calculate: The value of limit limx→∞exx4 by the use of L’Hospital’s Rule if necessary.

The provided function is limx→∞exx4.

Formula used:

L’Hospital’s Rule: If the limit of f(x)/g(x) when x approaches c from right produces indeterminate form 0/0, then limx→cf(x)g(x)=limx→cf′(x)g′(x).

Power rule: ddxxn=nxn−1.

Differentiation formula: ddx(ex)=ex.

Calculation:

Consider the function exx4.

Apply the limit x→∞ to the function exx4.

Substitute ∞ for x.

limx→∞exx4=e∞∞4=∞∞

As the direct substitution of limit produces indeterminate form ∞∞, so L’Hospital’s Rule can be applied.

Recall the L’Hospital’s Rule, limx→cf(x)g(x)=limx→cf′(x)g′(x)

Apply the L’Hospital’s Rule,

limx→∞exx4=limx→∞ddx(ex)ddx(x4)

Recall the power rule,

And the differentiation formula,

Apply the power rule and differentiation formula,

limx→∞exx4=limx→∞ddx(ex)ddx(x4)=limx→∞ex4⋅x4−1=limx→∞ex4x3

Now, again apply the limit x→∞.

Substitute ∞ for x.

limx→∞exx4=limx→∞ex4x3=e∞4∞3=∞∞

As the direct substitution of limit produces indeterminate form ∞/∞, so L’Hospital’s Rule can be applied.

Recall the L’Hospital’s Rule, limx→cf(x)g(x)=limx→cf′(x)g′(x)

Apply the L’Hospital’s Rule,

limx→∞ex4x3=limx→∞ddx(ex)ddx(4x3)

Recall the power rule,

And the differentiation formula,

Apply the power rule and differentiation formula,

limx→∞ex4x3=limx→∞ddx(ex)ddx(4x3)=limx→∞ex4⋅3x2=limx→∞ex12x2

Now, again apply the limit x→∞

Check out a sample textbook solution.

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Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. lim x → ∞ e x x 4

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Calculus Calculus 10th Edition

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary.

lim x → ∞ e x x 4