Evaluate lim x→0
10x/√(9+x)-√(9-x)
To evaluate the limit of the expression as x approaches 0, we can simplify the expression and then substitute 0 into the simplified expression.
Let's start by simplifying the expression:
lim x→0 (10x/√(9+x) - √(9-x))
First, we simplify the expression in the numerator:
lim x→0 (10x/√(9+x))
The square root in the denominator would result in an indeterminate form (0/0) when x = 0. To further simplify, we can multiply both the numerator and denominator by the conjugate of the denominator:
lim x→0 (10x * (√(9+x)+√(9+x))) / ((√(9+x) - √(9+x)) * (√(9+x) + √(9+x))))
This simplifies to:
lim x→0 (10x * (√(9+x)+√(9+x))) / ((9+x) - (9+x))
Now, we can simplify the expression even further:
lim x→0 (10x * 2√(9+x)) / (0)
Since the denominator is 0, we have an indeterminate form (0/0). To evaluate this limit, we need to use L'Hôpital's Rule. To do this, let's first find the derivative of the numerator and denominator with respect to x:
The derivative of the numerator is:
20 * √(9+x) + 10x * (1/2)(1/√(9+x))
The derivative of the denominator is:
1
Now, let's take the limit again using L'Hôpital's Rule:
lim x→0 (20 * √(9+x) + 10x * (1/2)(1/√(9+x))) / (1)
Substituting x = 0:
20 * √(9+0) + 10(0) * (1/2)(1/√(9+0))
20 * √9 + 0 * (1/2)(1/√9)
20 * 3 + 0 * (1/2)(1/3)
60
Therefore, the limit of the expression as x approaches 0 is equal to 60.