4. Applications of Derivatives
Differentials
Problem 4.7.19
Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 (x² + 2x) / (x +3)
Verified step by step guidance1
First, substitute x = 1 into the limit expression to check if it results in an indeterminate form. Calculate (1² + 2(1)) / (1 + 3).
If the result is not an indeterminate form, you can directly evaluate the limit. If it is an indeterminate form (like 0/0 or ∞/∞), proceed to the next step.
Since the expression is a rational function, apply l'Hôpital's Rule by differentiating the numerator and the denominator separately.
Differentiate the numerator: f(x) = x² + 2x, so f'(x) = 2x + 2. Differentiate the denominator: g(x) = x + 3, so g'(x) = 1.
Now, re-evaluate the limit using the derivatives: lim_x→1 (2x + 2) / 1, and substitute x = 1 to find the limit.
Was this helpful?
