17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (e¹/ₓ - 1)/(1/x)

Accepted Answer

Welcome back, everyone. Apply Laptal's rule to find the limit of the following function as x approaches infinity. Each of X equals e to the power of 2 divided by X minus 1 divided by 2 divided by X. We're given for answer choices A says 0, B 1/2, C1 and D 2. So let's write down the limit. We have limit as x approaches infinity. Of e to the power of 2 divided by X minus 1 divided by. 2 divided by X. As always, we begin with direct substitution. Now if Xs approaches infinity. Then we're going to get E to the power of 2 divided by infinity, which gives us 0, right? Essentially 2 divided by X is going to approach 0. We're subtracting 1, and in the denominator, we're taking 2 divided by infinity, right? So 2 divided by infinity, while this term is approaching 0. Because we're dividing 2 by an infinitely large number. So E to the power of 0 is 1 and 1 minus 1 gives us 0. Meaning overall we have an indeterminate form 0 divided by 0. And this means that we can apply laptop's rule for this form. If we want to apply a laptal's rule, we have to differentiate the numerator, so e to the power of. 2 divided by X minus 1, we're going to take the derivative of that term and we separately want to differentiate 2 divided by X in the denominator. When we perform differentiation, we're going to apply the direct substitution once again, right? To see if we eventually get a finite limit. So in this case, we're getting limit as X approaches infinity. Let's recall that the derivative of e to the power of axis eats the power of x. So this gives us eats the power of 2 divided by x. And now because our exponent consists of a function, well, we need to apply the chain rule, we essentially need to multiply by the derivative of 2 divided by X, and the derivative of -1 is 0. Now for the denominator, we are evaluating the derivative of 2 divided by X. And without evaluating that derivative we can notice that these two derivatives cancel each other out. Now from here, all that we have to do is simply evaluate the limit as X approaches infinity of E to the power of 2 divided by X. Let's attempt the direct substitution. This gives us e to the power of 2 divided by infinity. And we know that if we divide a number by infinity, the limit will be equal to 0, right? That term approaches 0. And this means that our answer is eat to the power of 0, which is simply 1. Now this is a finite value, so we have successfully arrived at the final answer, which corresponds to the answer choice C1. Thank you for watching.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (e¹/ₓ - 1)/(1/x)

Key Concepts

Limits

l'Hôpital's Rule

Exponential Functions

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