4. Applications of Derivatives
Differentials
Problem 4.7.28
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (x - 3 √x) / (x - √x)
Verified step by step guidance1
First, substitute x = 0 into the limit expression to check if it results in an indeterminate form. You will find that both the numerator and denominator approach 0, indicating that l'Hôpital's Rule may be applicable.
Next, apply l'Hôpital's Rule, which states that if the limit results in an indeterminate form of type 0/0, you can take the derivative of the numerator and the derivative of the denominator separately.
Differentiate the numerator, which is (x - 3√x). The derivative will involve using the power rule and the chain rule for the square root term.
Differentiate the denominator, which is (x - √x). Again, apply the power rule and the chain rule to find the derivative of the square root term.
Finally, substitute x = 0 into the new limit expression formed by the derivatives of the numerator and denominator to evaluate the limit.
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