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→ ∞ (e³ˣ ) / (3e³ˣ + 5)

Accepted Answer

Welcome back, everyone. Apply Laptal's rule to find the limit of the following function as X approaches infinity. F of X is equal to 7 e to the power of 2X divided by 21 E to the power of 2X plus 11. We're given the four answer choices A says 3 divided by 7, B 1 divided by 7, C 1 divided by 2, and D 1 divided by 3. So let's write down the limit. We have limit as X approaches infinity. Of 7E to the power of 2X divided by 21 E to the power of 2X plus E11. Let's begin with the direct substitution. If x approaches infinity, then e to the power of 2 X is going to be eats the power of infinity, and that term will approach infinity, right? We're going to get infinity in the numerator and in the denominator we have the same exponential term plus 11 while that's still infinity, right? And we have an indeterminate form infinity divided by infinity which allows us to apply Lapito's rule, and the rule says that we can take the derivative of the numerator and the denominator. So let's differentiate the numerator 7 eats the power of 2 X, and let's differentiate the denominator 21 eats the power of 2 x plus 11. We're going to get limit as X approaches infinity. First of all, let's factor out the constant, which is 7. Let's recall that the derivative of E to the power of X is, E to the power of X. So in this case, we're going to get E to the power of 2 X. But because our exponent consists of an inner function, we have to apply the chain rules, so we're multiplying by the derivative of 2 X. For the denominator, we're going to get 21 as our constant multiplied by e to the power of 2 X, and once again we're applying the chain rules, so we're multiplying by the derivative of 2 X. The derivative of 11 is 0 because it's a constant. And now notice that we can cancel out each the power of 2 X on both sides, and we can cancel out the derivative of 2 Xs on both sides. Which essentially gives us 7 divided by 21. This is now a finite number. And we can simplify it which gives us 1/3. So the correct answer to this problem corresponds to the answer choice D. Thank you for watching.

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

lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)

Key Concepts

Limits

L'Hôpital's Rule

Exponential Functions

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