4. Applications of Derivatives
Differentials
Problem 64
Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (x - √(x²+4x))
Verified step by step guidance1
Identify the limit to evaluate: lim_{x→∞} (x - √(x² + 4x)).
Substitute x with ∞ to check the form of the limit: this results in ∞ - ∞, which is an indeterminate form.
Rewrite the expression to facilitate the application of l'Hôpital's Rule: factor out x from the square root, giving lim_{x→∞} (x - x√(1 + 4/x)).
Simplify the expression to lim_{x→∞} x(1 - √(1 + 4/x)).
As x approaches ∞, apply l'Hôpital's Rule by differentiating the numerator and denominator if necessary, or simplify further to evaluate the limit.
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