$\text { Evaluate the limit } \lim _ { x \rightarrow 8 } \frac { - 2 x ^ { 2 } + 25 - 72 } { x - 8 } \text { first by using techniques from Chapter } 1$ then by using L'Hopital's Rule.
A) 8
B) 0
C) $- 7$
D) 7
E) does not exist

Question

$\text { Evaluate the limit } \lim _ { x \rightarrow 8 } \frac { - 2 x ^ { 2 } + 25 - 72 } { x - 8 } \text { first by using techniques from Chapter } 1$ then by using L'Hopital's Rule. 
A) 8
B) 0
C) $- 7$
D) 7
E) does not exist

Quizplus · Accepted Answer

First, by simplifying the expression: $\frac{-2x^2 + 25 - 72}{x - 8} = \frac{-2x^2 - 47}{x - 8}$. Factoring the numerator or using polynomial division directly doesn't easily simplify this expression due to the lack of a clear factor that cancels with $x - 8$. However, evaluating the limit using L'Hopital's Rule, we differentiate the numerator and the denominator separately. The derivative of the numerator is $-4x$, and the derivative of the denominator is 1. Evaluating the derivative of the numerator at $x = 8$ gives $-4(8) = -32$. Since L'Hopital's Rule applies to the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, and the original expression doesn't simplify directly to these forms without further manipulation, the direct application of L'Hopital's Rule was incorrectly described. The correct approach without L'Hopital involves simplifying and factoring if possible, but in this case, evaluating directly at $x = 8$ after simplification or recognizing the mistake in the initial approach to L'Hopital's Rule. The correct answer involves recognizing the error in applying L'Hopital's Rule directly without first ensuring the expression is in an indeterminate form that can be simplified or factored to apply the rule correctly. The correct evaluation involves simplifying the expression and then applying limits directly or finding a way to factor or cancel terms appropriately, which was not correctly described in the initial explanation. The mistake was in the direct application of L'Hopital's Rule without the correct preliminary steps, and the correct method involves more direct algebraic manipulation or correctly applying limit laws.

Then by Using L'Hopital's Rule −7- 7−7 D) 7 E) Does Not Exist

Then by Using L'Hopital's Rule $- 7$ D) 7
E) Does Not Exist