Find the limit (if it exists). (If an answer does not exist, enter DNE.)
lim_(x->59)(sqrt(x+5)-8)/(x-59)
To find the limit of the given expression as x approaches 59, we can directly substitute the value of 59 into the expression. However, plugging in the value of 59 into the expression would result in division by zero, which is undefined.
To overcome this issue, we can use algebraic manipulation to simplify the expression, which might help us identify a pattern or factor out common terms that could cancel out the denominator.
Let's simplify the expression step by step:
lim_(x->59) (sqrt(x+5) - 8) / (x - 59)
First, let's rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator:
lim_(x->59) [(sqrt(x+5) - 8) * (sqrt(x+5) + 8)] / [(x - 59) * (sqrt(x+5) + 8)]
Simplifying the numerator:
[ (x+5) - 8^2 ] / (x - 59) * (sqrt(x+5) + 8)
[x + 5 - 64] / (x - 59) * (sqrt(x+5) + 8)
[x - 59] / (x - 59) * (sqrt(x+5) + 8)
The (x - 59) terms in the numerator and denominator cancels out, leaving us with:
sqrt(x+5) + 8
Now let's plug in the value of 59 into this expression to find the limit:
sqrt(59+5) + 8 = sqrt(64) + 8 = 8 + 8 = 16
Therefore, the limit of the given expression as x approaches 59 is 16.