Two methods Evaluate the following limits in two different way...

Question

Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.

lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the limit using Lahapatal's rule. The limit as X approaches 0 of E to the power of 4 X plus 3 multiplied by E to the power of 2 X. -4, all divided by 1 minus E to the power of 4 X. Awesome. So it appears for this particular problem, we're asked to figure out what this evaluated limit is. So we're trying to evaluate this limit that is given to us, and that is our final answer that we're ultimately trying to solve for. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is -5 divided by 2, B is -2. C is 2 divided by 5 and D is 5. So, In order to evaluate this particular limit or any limit that you're asked to solve for using Lahapatal's rule, in order to recall and use Lahapatal's rule, as you should note, that we cannot use Lahapatal's rule unless We need unless we can verify that the limit is in an indeterminate form. So what does that mean? So an indeterminate form means that the limit yields an answer of 0 divided by 0 or infinity divided by infinity. So in order to evaluate a limit using Lahapato's rule, as we should recall, it has to mean when we use direct substitution, when we where it says the limit approach as X approaches whatever and this number that it's approaching, that's the value that we use for direct substitution. So in this case, X is approaching 0, so we need to plug in a value of 0 directly into our equation. And plug in a value of 0 in for all of our values of X. And if we get an answer that's in an indeterminate form as 0 divided by 0 or infinity divided by infinity, then we can use Lahapatal's rule. So, with that said, let's plug in a value of 0 in for all of our values of X, and when we do that, we'll find that E to the power of 4 multiplied by 0, plus. 3 Multiplied by E to the power of 2, multiplied by 0. -4, all divided by 1 minus E to the power of 4 multiplied by 0. Fantastic. So when we plug that into our calculator, we will find that we get an indeterminate form answer of 0 divided by 0. Fantastic. So that means we can safely use Lahapa's rule, and the expression once again is indeed in an indeterminate form. So therefore we can recall and apply the law hapatal's rule. And as we should recall a note, this law states that the limit as X approaches C of F of X divided by G of X. is an indeterminate form, then we could say that the limit as X approaches C of F of X divided by G of X is equal to the limit as X approaches C of F of X divided by G of X. And what does this prime mean? This prime means it's the first derivative. So that means that the limit as X approaches C of F of X divided by G of X is equal to the limit as X approaches C of the first derivative of FFX divided by the first derivative of G of X. Fantastic. So that means that if the limit on the right exists. Awesome. So now, let us find the derivatives of the numerator, which is F of X in this case, and the denominator, which is G of X. So we need to figure out in our original equation, our numerator, the value in the top is going to be E to the power of 4 X. Plus 3 multiplied by e 2X minus 4. So let us find the derivative of the numerator. So we will state that the numerator is F of X is equal to once again E 4X plus 3 multiplied by E of 2X minus 4. And when we take the first derivative, we will find that F of X is equal to 4 multiplied by E to the power of 4 X plus 6 multiplied by E to the power of 2 X. Fantastic. And now we need to find the derivative of the denominator, which is gonna be denoted as G of X and G of X is once again going to be equal to 1 minus E to the power of 4 X. Fantastic. And the first derivative of G of X is going to be equal to -4 multiplied by E to the power of 4 X. Fantastic. So now our next step is we can now apply law. Apital's rule, so you can write that the limit as X approaches 0 of 4 multiplied by the power of 4 X plus 6 multiplied by E to the power of 2 X, and we need to divide this by our denominator, which our denominator is -4 or 4 multiplied by E to the power of 4. X Awesome. Which we could say that this is also equal to the limit, as X approaches 0 of -1 minus 6, multiplied by E to the power of 2X divided by 4 multiplied by E to the power of 4 X. fantastic. So now we need to simplify a little bit more, and when we do, we will find that the limit as X approaches 0 of -1 minus 3 halves multiplied by E to the power of -2 X. So now, All we need to do is substitute in our value of X equals 0 because it's X approaches 0. So when we do that, you will find that -1 minus 3 halves or 3 divided by 2 multiplied by E to the power of 0. Which is, since it was negative, or if you want to be more precise so you don't get confused, you could write -1 minus 3 divided by 2 multiplied by E to the power of -2 multiplied by 0, which will give us the power of 0. It's going to be equal to when we plug that expression into our calculator, -5. Divided by 2 or 5 or negative 5 halves. And that's it. That is our final answer. Hooray, we did it. And this corresponds with multiple choice answer letter A, which once again states that our final answer is -5 divided by 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.
lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)

Key Concepts

Limits

L'Hôpital's Rule

Exponential Functions

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