Evaluate the limit, if it exists.
lim (sqrt(x+2) -3)/(x-7)
(x -> 7)
I tried multiplying the numberator and denominator by the conjugate of the numberator (sqrt(x+2) +3) but, unless I multiplied wrong, I still ended up getting 0/0, when my calculator and the book say that I should get an answer of 1/6. Is this the wrong way to approach the problem?
You multiplied wrong.
You should have gotten in the numberator
x+2-9= x-7, which divides out the x-7 in the denominator, leaveing (sqrt(x+2) + 3), or 1/6
Ah, ok. I multiplied the (x-7) out in the denominator and must have multiplied wrong. Thank you!
No, multiplying the numerator and denominator by the conjugate is the correct approach for this type of problem. However, it seems there might have been an error in your calculation. Let's go through the steps again to confirm.
To evaluate the limit, we can multiply the numerator and denominator by the conjugate of the numerator, which in this case is (√(x+2) + 3).
The expression becomes:
lim [(√(x+2) - 3)/(x-7)] * [(√(x+2) + 3)/(√(x+2) + 3)]
Simplifying the numerator using the difference of squares, we get:
lim [(x + 2) - 9]/[(x-7)(√(x+2) + 3)]
Further simplifying, we have:
lim (x - 7)/[(x-7)(√(x+2) + 3)]
Now, we can cancel out the common factor of (x-7) in the numerator and denominator.
The expression becomes:
lim 1/ (√(x+2) + 3)
Now, substitute x = 7 into the expression:
1/ (√(7+2) + 3)
1/ (√9 + 3)
1/ (3 + 3)
1/6
Therefore, the limit evaluates to 1/6, which confirms the answer given by your calculator and the book.
To evaluate this limit, it looks like you're on the right track by using the conjugate to simplify the expression. Let's go over the steps together to see if we can identify where things might have gone wrong.
First, let's rewrite the expression by multiplying both the numerator and the denominator by the conjugate of the numerator:
lim (sqrt(x+2) - 3)/(x - 7)
Multiply the numerator and the denominator by the conjugate of the numerator, which is (sqrt(x+2) + 3):
lim [(sqrt(x+2) - 3)/(x - 7)] * [(sqrt(x+2) + 3)/(sqrt(x+2) + 3)]
Simplifying this expression, you'll get:
lim [(sqrt(x+2))^2 - 3^2]/[(x - 7)(sqrt(x+2) + 3)]
Now, simplify further:
lim [x + 2 - 9]/[(x - 7)(sqrt(x+2) + 3)]
lim [x - 7]/[(x - 7)(sqrt(x+2) + 3)]
Notice that the (x - 7) terms will cancel out:
lim [1]/[sqrt(x+2) + 3]
Finally, substitute x = 7 into the simplified expression:
[1]/[sqrt(7+2) + 3]
[1]/[3 + 3]
[1]/[6]
1/6
Therefore, the limit as x approaches 7 is indeed equal to 1/6, as you stated.
It's possible that there was a mistake made in your calculations at some point. It's always helpful to double-check the algebraic steps and calculations to ensure accuracy.
I hope this helps clarify the problem-solving process! Let me know if you have any further questions.