Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→9 √x − 3 / x − 9

검증된 단계별 안내

Recognize that the limit \( \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} \) is an indeterminate form \( \frac{0}{0} \).

To resolve the indeterminate form, use algebraic manipulation. Multiply the numerator and the denominator by the conjugate of the numerator: \( \sqrt{x} + 3 \).

This gives: \( \lim_{x \to 9} \frac{(\sqrt{x} - 3)(\sqrt{x} + 3)}{(x - 9)(\sqrt{x} + 3)} \).

Simplify the expression: the numerator becomes \( x - 9 \) because \((\sqrt{x} - 3)(\sqrt{x} + 3) = x - 9\).

Cancel \( x - 9 \) from the numerator and denominator, resulting in \( \lim_{x \to 9} \frac{1}{\sqrt{x} + 3} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of a function as x approaches 9.

Indeterminate Forms

Indeterminate forms occur in calculus when direct substitution in a limit leads to an ambiguous result, such as 0/0. In the given limit, substituting x = 9 results in this form, necessitating further analysis, such as algebraic manipulation or applying L'Hôpital's Rule to resolve the limit.

Rationalizing the Numerator

Rationalizing the numerator is a technique used to simplify expressions involving square roots. By multiplying the numerator and denominator by the conjugate of the numerator, we can eliminate the square root and simplify the limit calculation. This method is particularly useful when dealing with limits that yield indeterminate forms.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.lim x→9 √x − 3 / x − 9