Find the horizontal asymptotes of each function using limits a...

Question

Find the horizontal asymptotes of each function using limits at infinity.

f(x) = (2e^x + 3) / (e^x + 1)

Accepted Answer

Welcome back every once another video. Identify any horizontal asymptotes of the function F of X is equal to 7E the power of x + 8 divided by e to the power of X + 2 by evaluating its limit as x approaches positive or negative infinity. Were given 4 answer choices A says y equals 4 and y equals 7. B says y equals 7 halves. C Y equals 0, and y equals 4, and d no horizontal asymptotes. So let's begin with the first limit. We're going to evaluate the limit as x approaches positive infinity of that function, and our function is 7 E. To the power of X + 8 divided by E to the power of X + 2. If we substitute infinity, well, essentially we're going to get infinity divided by infinity which is an indeterminate form, and this means that we want to simplify this function. And we can do that by dividing all of our terms by e to the power of X, and let's see why this is useful. Limit as X approaches infinity. Now dividing each term by eats the power of X, we're going to get 7 + 8 divided by e to the power of X, and in the denominator we're getting 1 + 2 divided by e to the power of X. And now E to the power of x approaches infinity when X approaches infinity. So the fractional terms now approach 0, right? Because we have a number divided by an infinitely large number. So, in this case, we're getting 7 divided by 1. Which is equal to 7. So our first horizontal asymptote is Y equals 7. Now for the second one, we want to evaluate the limit as X approaches negative infinity, and we have 78 to the power of X plus 8. Divided by e to the power of X + 2. Once again, we begin with direct substitution. We get 78 to the power of negative infinity plus 8. Divided by e to the power of negative infinity plus 2. So now the exponential terms approach 0. Right, because he is raised to an infinitely small number. And we are left with 8 divided by 2. Which gives us 4. So our second horizontal asymptote is Y equals 4. And if we look at the answer choices, we can see that the correct answer choice is option A, Y equals 4, and Y equals 7. Thank you for watching.

Find the horizontal asymptotes of each function using limits at infinity.
f(x) = (2e^x + 3) / (e^x + 1)

Key Concepts

Horizontal Asymptotes

Limits at Infinity

Exponential Functions

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