lim_(x->59)(sqrt(x+5)-8)/(x-59)??
note that when we substitute 59 to the expression in numerator and denominator,
(sqrt(x+5)-8)/(x-59)
(sqrt(59+5)-8)/(59-59)
(sqrt(64)-8)/0
8-8/0
0/0
thus we use L'hopital's Rule: we separately get the derivative of numerator and denominator. thus,
(num) = (sqrt(x+5)-8)
(num) = (x+5)^(1/2) - 8
d(num) = (1/2)*(x+5)^(-1/2)
d(num) = 1/(2*sqrt(x+5))
(deno) = x-59
d(deno) = 1
and we rewrite it again.
lim as x->59 of 1/(2*sqrt(x+5)) / 1
we can now substitute 59 to x:
1/(2*sqrt(59+5))
1/(2*sqrt(64))
1/(2*8)
1/16
hope this helps~ :)
To solve this limit, we can try using direct substitution by plugging in the value of 59 for x. However, if we do that, we end up with an indeterminate form of 0/0. Indeterminate forms occur when we cannot directly evaluate the limit by substitution.
To overcome this, we can simplify the expression by rationalizing the numerator. We can multiply both the numerator and denominator by the conjugate of the numerator, which is √(x+5) + 8. This will eliminate the square root in the numerator.
So, multiply both the numerator and denominator by (√(x+5) + 8):
lim(x->59) [(√(x+5) - 8) / (x-59)] * [(√(x+5) + 8) / (√(x+5) + 8)]
Now, we can use the property (a^2 - b^2) = (a + b)(a - b) to simplify the numerator. In our case, a = √(x+5) and b = 8:
lim(x->59) [(√(x+5))^2 - 8^2) / (x-59)] * [(√(x+5) + 8) / (√(x+5) + 8)]
This simplifies to:
lim(x->59) [(x + 5 - 64) / (x-59)] * [(√(x+5) + 8) / (√(x+5) + 8)]
Further simplifying:
lim(x->59) [(x - 59) / (x-59)] * [(√(x+5) + 8) / (√(x+5) + 8)]
Since (x - 59) / (x - 59) equals 1 (except when x = 59, but we are evaluating the limit as x approaches 59, so it doesn't affect the final result), we can simplify it further:
lim(x->59) [(√(x+5) + 8) / (√(x+5) + 8)]
Now, we can substitute x = 59 into the expression:
[(√(59+5) + 8) / (√(59+5) + 8)]
This simplifies to:
[(√64 + 8) / (√64 + 8)]
Using direct substitution, we find that the square root of 64 is 8:
[(8 + 8) / (8 + 8)]
Simplifying the numerator and the denominator:
[16 / 16]
Finally, we get:
1
Therefore, the limit of (sqrt(x+5) - 8) / (x-59) as x approaches 59 is equal to 1.