Evaluate lim_x→2 (x³ - 3x² + 2) / (x-2) using l’Hôpi...

Question

Evaluate lim_x→2 (x³ - 3x² + 2) / (x-2) using l’Hôpital’s Rule and then check your work by evaluating the limit using an appropriate Chapter 2 method.

Accepted Answer

Welcome back, everyone. Evaluate the limit. Use Lapital's rule if necessary. We want to evaluate the limit as x approaches 3 of X cubed minus 8 X2 + 15 x divided by X minus 3, and where given 4 answer choices AS-6, B -3, C0, and D 4. So first of all, let's identify the limit. It says limit as X approaches 3 of X cubed minus 8 X2 + 15 X divided by X minus 3. So what we want to do is simply attempt direct substitution. Now let's substitute 3 into the given expression. So we get 3 cubed minus 8 multiplied by 3 squad plus 15 multiplied by 3 divided by 3 minus 3. Now, what we get in the numerator, so 27 minus 72. Plus 45, so that's 0, and overall we get an indeterminate form which is 0 divided by 0, and for this form we can use the Lapito's rule. So, The rule tells us that we can see the derivative of the top function and the bottom function and then attempt the direct substitution once again. We can continue the process until we get rid of the indeterminate forms and end up with an actual limit. So what we're going to do is simply take the derivative of the top function. We're going to say that we're differentiating X cubed minus 8 X2 + 15 x for the numerator, and we're going to take the derivative of the denominator. So the derivative of X minus 3, and this is going to be equal to the same limit, right? So we get limit as X approaches 3 of 3 x 2. Minus 16 X + 15, and now the derivative in the denominator is simply 1, right? Because the derivative of X is 1, the derivative of -3 is 0. And now once again let's attempt direct substitution. It is going to be successful because the denominator has 1, right? So we got 3 multiplied by 3 squad minus 16 multiplied by 3 + 15. And now let's perform the calculation. So we get 3 cubed, which is 27 minus 48 + 15. So 27 + 15 gives us 42 and 42 minus 48 gives us -6. So the correct answer to this problem corresponds to the answer choice A. Thank you for watching.

Evaluate lim_x→2 (x³ - 3x² + 2) / (x-2) using l’Hôpital’s Rule and then check your work by evaluating the limit using an appropriate Chapter 2 method.

Key Concepts

L'Hôpital's Rule

Polynomial Functions

Limit Evaluation Techniques

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