109-112 {Use of Tech} Calculating limits The following limits ...

Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\lim_{x \to 2} \frac{{(x^{2} - 3)}^{5} - 1}{x - 2}$

Accepted Answer

Hello 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. Apply Lahapatal's rule to find the limit of the following function as X approaches -1. H X is equal to x 3 + 2 to the power of 7 minus 1 divided by X + 1. OK. So it appears for this particular problem we're asked to figure out what the limit is of this specific function as X approaches -1 using Lahapital's rule. Awesome. So now that we know that we're trying to figure out the limit of this function, let us read off our multiple choice answers to see what our final answer might be. A is 7, B is 21, C is -1. And D is -3. OK. So, the limit that we're trying to evaluate, as we should note, is we're trying to figure out the limit as X approaches -1 of X to the power of 3 + 2. All to the power of 7 minus 1 all divided by X + 1. OK, so we need to recognize at this point, as we should recall, that the direct substitution of X equals -1 leads to an indeterminate form which indicates the potential use of Lajapatal's rule. So, as we should notice, the limit as X approaches -1 or X to the power of 3. Plus 2 alter the power of 7 minus 1 all divided by X plus 1. is equal to, and don't forget your bracket, when we plug in our value for -1 and for X, so -1. Which is to the power of 3 + 2 to the power of 7. -1. All divided by -1 + 1. Awesome. So, when we plug this into our calculator, you will find that it's gonna be equal to 0 divided by 0. So when we apply, so when we recall and apply Lajapatal's rule, we need to differentiate the numerator, which is the top part of our division bar, and the denominator, which is the bottom part of our division bar separately with. to X. So for the numerator, we're gonna need to recall and use the chain rule. So when we do that, we will find that the limit as X approaches -1 of D by DX. Of X to the power of 3 + 2, all to the 7 minus 1. Divided by D by DX of X + 1. And this will be equal to the limit as X approaches -1 of 7 multiplied by X 3 + 26 multiplied by 3 x 2. Divided by 1, and this will be equal to the limit as X approaches -1 of 21 X 2 multiplied by X 3 + 2 to the power of 6. So now our next step is we need to evaluate the limit as X approaches -1, so now we're evaluating. So we need to substitute in our value for X equals -1 into our simplified function. So when we do that, we will find that the limit as X approaches -1 of 21 X2 multiplied by X to the power of 3 + 2 to the power of 6. And we need to take this, and this will be equal to 21 multiplied by -1 to the power 2 multiplied by -1 to the power of 3 + 2 to the power of 6. So when we plug this into our calculator, we will find that it will yield an answer of 21, and that is our final answer. Hooray, we did it. So the limit is 21, and this corresponds with multiple choice answer letter B, which once again is 21. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.
b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.
$\lim_{x \to 2} \frac{{(x^{2} - 3)}^{5} - 1}{x - 2}$

Key Concepts

Limits

Chain Rule

Indeterminate Forms

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