(sqrt(9x+5)-sqrt(68))/(x-7) limit as x approaches 7
the answer i got was .0546 i am not sure if this is right also, if it is ...is there a more exact fraction .... like if the answer was 0.3333333 a better way to write it would be 1/3
Use L'Hopital's rule and take the ratio of the derivatives of numerator and denominator.
The derivative of the numerator at x = 7 is
(9/2)/(9x+5)^(1/2)= (9/2)/sqrt68 = 0.5457..
and the derivative of the denominator is 1.
Therefore your answer is correct. If you want the exact value, use
(9/2)/sqrt68
Thank you so much!!
Actually, your answer is ten times too low. Did you slip a decimal point somewhere?
i accidentally put the 0 in the wrong place in the above response i had 0.546...
To find the limit of the given expression as x approaches 7, we can simplify it further. Here's how:
First, let's simplify the expression:
(sqrt(9x+5) - sqrt(68))/(x-7)
We can simplify the two square roots:
(sqrt(9x+5) = sqrt(9) * sqrt(x+5) = 3 * sqrt(x+5)
(sqrt(68) = sqrt(4 * 17) = 2 * sqrt(17)
Now our expression becomes:
(3 * sqrt(x+5) - 2 * sqrt(17))/(x-7)
Next, let's multiply the numerator and denominator by the conjugate of (x-7). The conjugate of (x-7) is (x-7) itself.
((3 * sqrt(x+5) - 2 * sqrt(17))/(x-7)) * ((x-7)/(x-7))
= (3 * sqrt(x+5) - 2 * sqrt(17))(x-7)/(x-7)^2
Now we can simplify further by cancelling out (x-7) in both the numerator and denominator:
= (3 * sqrt(x+5) - 2 * sqrt(17))/(x-7)
= (3 * sqrt(x+5) - 2 * sqrt(17))/(x-7)
To find the limit as x approaches 7, we can evaluate the expression by substituting x=7 into the simplified expression:
= (3 * sqrt(7+5) - 2 * sqrt(17))/(7-7)
= (3 * sqrt(12) - 2 * sqrt(17))/0
Since the denominator is 0, it indicates that the limit does not exist in this case.
Based on the work you provided, it seems like you may have made a mistake in your calculation or simplified the expression differently. It is recommended to recheck your steps or use a calculator or software to evaluate the expression for a more precise result.