4. Applications of Derivatives
Differentials
Problem 17b
Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→2 (x² - 2x / (x² - 6x + 8)
Verified step by step guidance1
First, substitute x = 2 into the limit expression to check if it results in an indeterminate form. Calculate (2² - 2*2) / (2² - 6*2 + 8).
If the result is an indeterminate form like 0/0 or ∞/∞, apply l'Hôpital's Rule, which states that you can take the derivative of the numerator and the derivative of the denominator.
Differentiate the numerator, which is x² - 2x, to get 2x - 2.
Differentiate the denominator, which is x² - 6x + 8, to get 2x - 6.
Now, substitute x = 2 into the new limit expression formed by the derivatives to evaluate the limit.
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