lim_(x->59)(sqrt(x+5)-8)/(x-59)??
To find the limit of the given expression as x approaches 59, we can use a direct substitution method. However, substituting 59 directly into the expression results in an indeterminate form of 0/0, which does not provide a clear answer. To overcome this, we can use algebraic manipulation.
Let's simplify the expression step by step:
First, let's rationalize the numerator. To do this, we can multiply both the numerator and the denominator by the conjugate of the numerator, which is sqrt(x+5)+8. This will eliminate the square root and simplify the expression.
lim(x->59) [(sqrt(x+5)-8)/(x-59)] * [(sqrt(x+5)+8)/(sqrt(x+5)+8)]
Now, we can simplify the denominator by applying the difference of squares formula, which is a^2 - b^2 = (a+b)(a-b). In this case, a = sqrt(x+5) and b = 8.
lim(x->59) [((sqrt(x+5))^2 - 8^2)] / [(x-59)(sqrt(x+5)+8)]
Simplifying further:
lim(x->59) [(x+5 - 64)] / [(x-59)(sqrt(x+5)+8)]
lim(x->59) [x - 59] / [(x-59)(sqrt(x+5)+8)]
Now, we can cancel out the common factor (x-59) from the numerator and the denominator:
lim(x->59) 1 / (sqrt(x+5)+8)
Finally, we can substitute x = 59 into the expression, which results in:
1 / (sqrt(59+5) + 8)
1 / (sqrt(64) + 8)
1 / (8 + 8)
1 / 16
0.0625
Therefore, the limit of the given expression as x approaches 59 is 0.0625.