Question

Find $\lim _{x \rightarrow 2} \frac{2-\sqrt{x+2}}{x-2}$.

   Find $\lim _{x \rightarrow 2} \frac{2-\sqrt{x+2}}{x-2}$.

Basic Technical Mathematics with Calculus

Basic Technical Mathematics with Calculus

Allyn J.Washington,… 12th Edition

Chapter 23, Problem 27

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Solved by Expert Carson Merrill

08/03/2021

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Find $\lim _{x \rightarrow 2} \frac{2-\sqrt{x+2}}{x-2}$.

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Key Concepts

Indeterminate Forms

Indeterminate forms, such as the 0/0 form, occur when both the numerator and the denominator of a rational expression approach zero. Recognizing an indeterminate form is essential for applying appropriate techniques, such as algebraic manipulation or L’Hôpital’s Rule, to evaluate the limit.

Rationalization

Rationalization is an algebraic technique used to eliminate radicals from the numerator or denominator of an expression. By multiplying by the conjugate, the expression is simplified, which can help resolve indeterminate forms and facilitate the evaluation of limits.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ?/?. It involves taking the derivative of the numerator and the derivative of the denominator separately and then obtaining the limit of the resulting quotient. This rule is applicable when the functions are differentiable near the point of interest.

Limits

Limits involve determining the value that a function approaches as the input approaches a specific point. They are fundamental in calculus for understanding the behavior of functions as their arguments get arbitrarily close to a certain value, even when the function may not be explicitly defined at that point.

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