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

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Homework Help Overview

The discussion revolves around finding the limit of the expression (sqrt(x + 2) - 3)/(x - 7) as x approaches 7. Participants explore the implications of the function being undefined at x = 7 and the behavior of the numerator and denominator as they approach zero.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Some participants discuss the importance of limits beyond direct substitution, noting that both the numerator and denominator approach zero at x = 7. Others suggest rationalizing the numerator and multiplying by a conjugate to simplify the expression. There are questions about the correctness of methods used and the steps taken in the simplification process.

Discussion Status

The discussion is ongoing, with participants providing guidance on algebraic manipulation and expressing differing views on the validity of approaches. Some participants emphasize the need for careful simplification before substituting values, while others question the logic behind certain conclusions drawn about the limit's existence.

Contextual Notes

Participants note that the problem may not strictly adhere to typical homework guidelines, with some expressing frustration over the complexity of the limit compared to previous exercises they have encountered.

  • #31
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.
 

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