Limit of (sqrt(x + 2) - 3)/(x - 7) as x approaches 7

Thread starter Victor II
Start date Feb 7, 2014
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Limit

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SUMMARY

The limit of the expression (sqrt(x + 2) - 3)/(x - 7) as x approaches 7 is an indeterminate form of 0/0, which requires further analysis to evaluate. Users in the discussion emphasized the importance of rationalizing the numerator by multiplying both the numerator and denominator by (sqrt(x + 2) + 3) to simplify the expression. This method allows for cancellation of terms, leading to a clearer evaluation of the limit. The correct approach involves recognizing that while the function is undefined at x = 7, the limit can still exist through proper algebraic manipulation.

PREREQUISITES

Understanding of limits in calculus
Knowledge of rationalizing expressions
Familiarity with indeterminate forms, specifically 0/0
Basic algebraic manipulation skills

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Study the concept of limits and continuity in calculus
Learn techniques for resolving indeterminate forms, such as L'Hôpital's Rule
Practice rationalizing expressions in limit problems
Explore graphical methods for evaluating limits using graphing calculators

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Students studying calculus, educators teaching limit concepts, and anyone seeking to improve their problem-solving skills in evaluating limits and understanding continuity in functions.

Feb 9, 2014

Victor M. II

HallsofIvy said:

Have you not actually taken a course in limits? You should have learned that the whole point of "limits" is that they give us more subtle information than just plugging the values into the function. The fact that both numerator and denominator are 0 at at x= 7 tells us nothing about the limit. That depends on exactly how the numerator and denominator go to 0.

Here what you need to do is multiply both numerator and denominator by \sqrt{x+ 2}+ 3, then take the limit as x goes to 7.

Thank you. I solved the problem. Only introductory algebra is involved.