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

Question

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

f(x) = (3e^5x + 7e^6x) / (9e^5x + 14e^6x)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with limits. This problem says to identify any horizontal asymptotes of the function F of X, which is equal to the quantity of 5 multiplied by E raised to the 7 X, plus 3 multiplied by E raised to the 8 X in quantity, divided by the quantity of 25 multiplied by E raised to the 7 X, plus 6 multiplied by E raised to the 8 X by evaluating its limit as x approaches plus or minus infinity. We're given 4 possible choices as our answers. For choice A, we have Y is equal to 1 divided by 5, and Y is equal to 2. For choice B, we have Y is equal to 0, and Y is equal to 1 divided by 2. For choice C, we have Y is equal to 1 divided by 5, and Y is equal to 1 divided by 2. And for choice D, we have no horizontal asymptotes. Now we need to find the horizontal asymptotes for this function by evaluating its limit as X approaches plus or minus infinity. Let's start off by evaluating the limit as X approaches infinity. So we're looking at the limit. As X purchases infinity of the quantity of 5 multiplied by E raised to the 7 X. Plus 3 multiplied by E raised to the 8 X. In quantity, divided by the quantity of 25 multiplied by E to the 7 X. Plus 6 multiplied by E to the 8 X. Now, if we simply try to substitute in infinity for X into our equation, we will get infinity divided by infinity, which is an indeterminate form. So, in order to get rid of that indeterminant form, we first need to divide the numerator and denominator by E rays to the 8 X. So doing so, this will be equal to the limit. As X approaches infinity of the quantity of 5 multiplied by E to the 7 X, divided by E 8 X. Plus 3 multiplied by E to the 8 X. Divided by E to the 8X. In quantity, divided by the quantity of 25 multiplied by E to the 7 X. Divided by E to the 8 X. Plus, 6 multiplied by E to the 8 X. Divided by E to the 8 X. So we can simplify this expression, so this is, this will equal the limit. As X approaches infinity. Of the quantity For our first term will have 5 multiplied by E to the 7 X divided by E to the 8 X, that's gonna be 5 divided by E to the X. Then we'll have plus 3 multiplied by E to the index divided by E to the index, which is just going to give me 3, and that quantity is going to be divided by the quantity of 25 multiplied by E 7 X divided by E to the 8X, which is 25 divided by E to the X. Then we'll have plus 6 multiplied by E to the 8 X divided by ETD 8X, which is just 6. So at this point now we can actually evaluate the limit. So, ETD X approaches infinity as X approaches infinity. So, for our first term in the numerator, we'll have 5 divided by infinity, which is going to approach 0. So this quantity is gonna be equal to the quantity of 0 + 3 in the numerator, and that quantity is gonna be divided by And the denominator will have 25 divided by E to the X, which approaches 0 as X approaches infinity, that the denominator is approaching infinity, so that's gonna be 0. Plus dicks. So, we can now simplify this expression, and this is going to be equal to 1 divided by 2. And so that means our first asymptote is at Y is equal to 1 divided by 2. That's the first half of our answer. Now, for the second half, we need to take the limit as X purchase minus infinity. So we'll be looking at the limit. As X approaches minus infinity. Of the quantity of 5, multiplied by E to the 7 X, plus 3 multiplied by E to the 8 X. In quantity, divided by the quantity of 25, multiplied by E to the 7 X. Plus 6 multiplied by E to the 8 X. Now, if we substitute N minus infinity for X, we're actually going to get 0 divided by 0, which is an indeterminate form, since E raises to the minus infinity is approaching 0, so all of our terms will be zero. So, what we need to do to get it out of that indeterminant form is divide our numerator and denominator by E to the 7 X. So doing so, this is going to be equal to the limit. As X approaches minus infinity. Of the quantity of 5 multiplied by E to the 7 X divided by E 7 X. Plus 3 multiplied by E to the 8 X. Divided by E to the 7 X. In quantity, divided by the quantity of 25, multiplied by E to the 7 X. Divided by E to the 7 X. Plus 6 multiplied by E to the 8 X. Divided by E to the 7 X. So we can simplify that expression. And this is going to be equal to. The limit As X approaches minus infinity. Of the quantity in the numerator for the first term, we'll have multiplied by, or sorry, we'll have 5 multiplied by E to the 7 X divided by E to the 7 X, which gives me just 5. And we'll have plus For the next term we'll have 3 multiplied by E to the 8 X divided by E to the 7 X, that is just 3 multiplied by E to the X. That quantity is going to be divided by the quantity on the denominator, where you're going to have 25 multiplied by E to the 7 X divided by E to the 7X, which is just going to give you 25. Don't have what? 6 multiplied by E to the 8 X divided by E to the 7 X, which is 6 multiplied by E to the X. So now we can actually evaluate this limit. So this is going to be equal to. In the numerator, I want to take the limit as X approaches minus infinity. For the first time we're gonna have 5 since it's constant. Plus 3, multiplied by E to the X as X approaches a minus infinity, actually approaches 0. So we'll have plus 0 in the newer, and then that quantity is going to be divided by 25. Plus 0. Since the first term is a constant, so taking its limit, giving back just that constant, and then for the next term, the 6 multiplied by e to the X as X approaches in minus infinity, that term approaches 0. So we can simplify our expression here, and this is going to be equal to 1 divided by 5. So our next horizontal asymptote is that Y is equal to 1 divided by 5. So now that we have calculated both of our horizontal asymptotes, we can come back up to our answer choices, and the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Find the horizontal asymptotes of each function using limits at infinity.
f(x) = (3e^5x + 7e^6x) / (9e^5x + 14e^6x)

Key Concepts

Horizontal Asymptotes

Limits at Infinity

Exponential Functions

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