Math / Calculus

Question(1 point)
Evaluate the limit using L'Hospital's rule if necessary. $\lim _{x \rightarrow+\infty} \frac{x^{16}}{e^{x}}$
Answer: $\square$

Studdy Solution

STEP 1

1. We are given the limit $\lim_{x \rightarrow +\infty} \frac{x^{16}}{e^{x}}$ .
2. We need to evaluate this limit as $x$ approaches positive infinity.
3. L'Hospital's rule can be applied if the limit is in an indeterminate form like $\frac{\infty}{\infty}$ .

STEP 2

1. Determine if the limit is in an indeterminate form.
2. Apply L'Hospital's Rule if necessary.
3. Evaluate the limit after applying L'Hospital's Rule.
4. Conclude the value of the limit.

STEP 3

Check if the limit is in an indeterminate form:
As $x \rightarrow +\infty$ , both the numerator $x^{16}$ and the denominator $e^x$ approach $\infty$ .
Thus, the limit is in the form $\frac{\infty}{\infty}$ , which is an indeterminate form.

STEP 4

Apply L'Hospital's Rule:
L'Hospital's Rule states that if $\lim_{x \rightarrow c} \frac{f(x)}{g(x)}$ is in an indeterminate form, then:
$\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}$
Differentiate the numerator and the denominator:
Numerator: $\frac{d}{dx}[x^{16}] = 16x^{15}$
Denominator: $\frac{d}{dx}[e^x] = e^x$

STEP 5

Evaluate the new limit:
$\lim_{x \rightarrow +\infty} \frac{16x^{15}}{e^x}$
This is still in the form $\frac{\infty}{\infty}$ , so apply L'Hospital's Rule again.
Differentiate again:
Numerator: $\frac{d}{dx}[16x^{15}] = 240x^{14}$
Denominator: $\frac{d}{dx}[e^x] = e^x$
Evaluate the new limit:
$\lim_{x \rightarrow +\infty} \frac{240x^{14}}{e^x}$
Continue applying L'Hospital's Rule repeatedly until the power of $x$ in the numerator is zero.

STEP 6

Continue applying L'Hospital's Rule 16 times in total:
After 16 applications, the numerator becomes a constant (the derivative of $x^0$ ) and the denominator remains $e^x$ .
The final limit is:
$\lim_{x \rightarrow +\infty} \frac{16!}{e^x} = 0$
since $e^x$ grows much faster than any constant.
The value of the limit is:
$\boxed{0}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms