Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any ...

Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=3e^x+10 / e^x

Accepted Answer

Welcome back, everyone. Determine the horizontal asymptotes of the function G of X equals 17 eats the power of X plus 19 divided by e the power of X. We're given for answer chooses A Y equals 0, B Y equals 17, C Y equals -17, and D Y equals -17, and Y equals 17. So if we want to identify horizontal asymptotes, while we essentially want to evaluate two limits. The first one is limit as x approaches negative infinity of the function, and the second one, well, we want to approach positive infinity. And if we get finite numbers, those will be our horizontal asymptotes in the form of Y equals the value of the limit. So our function is 17 E to the power of X plus 19 divided by E to the power of X. What we're going to do before we evaluate the limit directly, well, we are going to simplify it, right? So let's split the given fraction into two separate fractions. We're going to get limit as X approaches negative infinity, 17 e to the power of X, divided by e to the power of X gives us 17, and then we end up with plus 19 divided by e to the power of X. And now let's try direct substitution. First of all, we get 17, it's independent of X plus 19 divided by it's the power of negative infinity, right? Now each the power of negative infinity can be written as 19 Multiplied by Eats the power of positive infinity. So essentially we rewrote one divided by eats the power of negative infinity as eats the power of positive infinity, we took the reciprocal of the value. And now let's notice that this gives us a limit of infinity. 17 is a number, 19 multiplied by eachs the power of infinity gives us infinity. So we do not have any horizontal asymptotes just yet because this is not a finite number. The next limit we are going to evaluate is limit as X approach is positive infinity, we're going to use our simplified form. 17 + 19 divided by e to the power of X and in this case while x approaches positive infinity, so we're going to get 17 + 19 divided by e to the power of positive infinity, and this means that Our denominator. Approaches an infinitely large number. So whenever we divide a number by an infinitely large number, we're going to get a limit of 0. So in this case, the limit is going to be equal to 17, meaning we only have one horizontal asymptote because the previous limit did not correspond to a finite number. So the only Horizontal asymptote that we have is Y equals 17. So the final answer to this problem is option B. Thank you for watching.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=3e^x+10 / e^x

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Exponential Functions

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