Evaluate

Question

Evaluate ${\displaystyle\lim_{x o\infty}{f(x)}}$ and ${\displaystyle\lim_{x o-\infty}{f(x)}}$ .

$f (x) = \frac{6 e^{x} + 20}{3 e^{x} + 4}$

Accepted Answer

Welcome back, everyone. Determine limits of FF X as X approaches positive and negative infinity for the function F of X equals 4 E to the power of X + 17 divided by 2 to the power of X + 3. Let's begin with the first one. We are going to start with limit as X approaches positive infinity, and we want to plug in 4E to the power of X + 17. And divide that by 2 e to the power of X plus 3. If we apply the right substitution, well, we're going to get an indeterminate form, infinity divided by infinity, and this means that we want to simplify it, and we can do that by dividing each term by e to the power of X. This gives us a limit as x approaches infinity. Now 4 E to the power of facts divided by E to the power of facts is just 4. Then we are dividing 17 by e to the power of X, and in the denominator we get 2 + 3 divided by e to the power of X. So we have divided each term by e to the power of X. And now we can see that as X approaches infinity, our denominator. For 17 divided by e to the power of X and 3 divided by e to the power of X, well, those approach is 0, right, because the denominator approaches infinity and a number divided by an infinitely large number has a limit of 0. So now we end up with 4 divided by 2, which gives us a finite number, and we have successfully evaluated the limit. 4 divided by 2 gives us 2, and that's our first limit. For the next one, we are evaluating limit as X approaches negative infinity of 4 eats the power of X plus 17 divided by. 2 E to the power of X + 3, and in this case we can apply the right substitution once again. So now 4 E to the power of negative infinity is exactly the same as 4 divided by E to the power of positive infinity, so we're using the reciprocal rule. In the denominator, we get 2 E to the power of negative infinity, which is 2 divided by E to the power of positive infinity, and then we're adding 3. And Similarly to the previous part, if we divide a number by an infinitely large number, those would be. Equal to 0 in terms of a limit, and we end up with two finite numbers, 17 and 3. Which means that we can now evaluate the limit and this gives us 17. Divided by 3. Since it is a finite number, we can, we can drop the brackets. Because this is our final answer. So the first limit as X approaches positive infinity is equal to 2, and the second one, as X approaches negative infinity is 17 divided by 3. Thank you for watching.

Evaluate ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and ${\displaystyle\lim_{x\to-\infty}{f(x)}}$ .

$f (x) = \frac{6 e^{x} + 20}{3 e^{x} + 4}$

Key Concepts

Limits at Infinity

Exponential Functions

Rational Functions

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