/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Consider the following applicati... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 40

Consider the following application of L'H么pital's Rule: $$ \lim _{x \rightarrow 1} \frac{3 x^{2}-2 x-1}{x^{2}-x}=\lim _{x \rightarrow 1} \frac{6 x-2}{2 x-1}=\lim _{x \rightarrow 1} \frac{6}{2}=3 $$ Is it correct? Justify.

Short Answer

Expert verified
The given application of L'H么pital's Rule is incorrect. After checking the conditions and applying L'H么pital's Rule, we find that \( \lim_{x \rightarrow 1} \frac{3x^2 - 2x - 1}{x^2 - x} = \lim_{x \rightarrow 1} \frac{6x - 2}{2x - 1} = 4 \), not 3.

Step by step solution

01

Checking the conditions for applying L'H么pital's Rule

To apply L'H么pital's Rule, two conditions must be met: \( \lim_{x \rightarrow 1} 3x^2 - 2x -1 = 0\) and \( \lim_{x \rightarrow 1} x^2 - x = 0\). Let's check these conditions: \( \lim_{x \rightarrow 1} 3x^2 - 2x - 1 = 3(1)^2 - 2(1) - 1 = 3 - 2 - 1 = 0\) \( \lim_{x \rightarrow 1} x^2 - x = (1)^{2} - (1) = 1 - 1 = 0\) Both conditions are met, so we can apply L'H么pital's Rule.
02

Applying L'H么pital's Rule

Differentiating numerator and denominator separately with respect to x: \( \frac{d}{dx}(3x^2 - 2x - 1) = 6x - 2 \) \( \frac{d}{dx}(x^2 - x) = 2x - 1 \) Now, using L'H么pital's Rule: \( \lim_{x \rightarrow 1} \frac{3x^2 - 2x - 1}{x^2 - x} = \lim_{x \rightarrow 1} \frac{6x - 2}{2x - 1} \)
03

Evaluating the limit

Now, we evaluate the limit: \( \lim_{x \rightarrow 1} \frac{6x - 2}{2x - 1} = \frac{6(1) - 2}{2(1) - 1} = \frac{6 - 2}{2 - 1} = \frac{4}{1} = 4 \)
04

Conclusion

So, the provided answer is incorrect. The correct limit after applying L'H么pital's Rule is: \( \lim_{x \rightarrow 1} \frac{3x^2 - 2x - 1}{x^2 - x} = 4\)

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Most popular questions from this chapter

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