Evaluate the limit, if it exists.
lim x->16 (4-sqrt(x))/(16x-x^2)
To evaluate the limit, we substitute 16 in place of x in the expression and simplify:
lim x->16 (4 - sqrt(x))/(16x - x^2) = (4 - sqrt(16))/(16(16) - 16^2)
= (4 - 4)/(256 - 256)
= 0/0
The resulting expression is of the form 0/0, which means that we can use L'Hospital's Rule to determine the limit. Taking the derivative of both the numerator and denominator:
lim x->16 (4 - sqrt(x))/(16x - x^2) = lim x->16 (-1/2sqrt(x))/(16 - 2x) / (16 - 2x)
= lim x->16 -1/[2sqrt(x)(16 - 2x)]
Now we substitute 16 back into the expression:
lim x->16 -1/[2sqrt(x)(16 - 2x)] = -1/[2sqrt(16)(16 - 2(16))]
= -1/[2(4)(16 - 32)]
= -1/[-64]
= 1/64
Therefore, the limit of the expression as x approaches 16 is 1/64.
To evaluate the limit of the given function, we can try to simplify the expression and then substitute the value of x.
First, let's simplify the expression by factoring out the common factor of (4 - √(x)) in the numerator:
lim x->16 (4 - √(x)) / (16x - x^2)
= lim x->16 (4 - √(x)) / (x(16 - x))
Next, let's simplify the expression further by rationalizing the denominator. To do this, we multiply both the numerator and denominator by the conjugate of the denominator, which is (16 - x):
lim x->16 [(4 - √(x)) / (x(16 - x))] * [(16 - x) / (16 - x)]
= lim x->16 (4 - √(x))(16 - x) / (x(16 - x)(16 - x))
= lim x->16 (4(16 - x) - √(x)(16 - x)) / (x(16 - x)(16 - x))
= lim x->16 (64 - 4x - √(x)(16 - x)) / (x(16 - x)(16 - x))
= lim x->16 (64 - 4x - √(x)(16 - x)) / (x^3 - 16x^2 + 16x^2 - 256x)
Now, we can substitute x = 16 into the expression:
lim x->16 (64 - 4x - √(x)(16 - x)) / (x^3 - 16x^2 + 16x^2 - 256x)
= (64 - 4(16) - √(16)(16 - 16)) / (16^3 - 16(16)^2 + 16(16)^2 - 256(16))
= (64 - 64 - √(16)(0)) / (4096 - 4096 + 4096 - 4096)
= 0 / 0
We end up with an indeterminate form of 0/0. This means we cannot evaluate the limit directly using substitution. To find the limit, we need to employ additional techniques such as L'Hopital's Rule, simplification, or further algebraic manipulation.
To evaluate the limit, we can plug in the value x = 16 into the expression (4 - sqrt(x))/(16x - x^2) and see what we get. However, before doing that, we must check if there are any factors that can be canceled out to simplify the expression.
Let's start by factoring the denominator 16x - x^2:
16x - x^2 = x(16 - x)
Now let's rewrite the limit expression with the factored denominator:
lim x->16 (4 - sqrt(x))/(x(16 - x))
Next, let's consider the numerator:
As x approaches 16, sqrt(x) will approach sqrt(16), which is 4. Therefore, the numerator (4 - sqrt(x)) will approach 0.
Now, let's consider the denominator:
As x approaches 16, (x(16 - x)) will approach 0*0, which is also 0.
So, we have a limit of the form 0/0, which is an indeterminate form. In this case, we can try to simplify the expression further or use other techniques to evaluate the limit.
Let's simplify the expression by factoring out a common factor of (4 - sqrt(x)) from the numerator:
lim x->16 (4 - sqrt(x))/(x(16 - x)) = lim x->16 (4 - sqrt(x))/(x(16 - x))
= lim x->16 1/(x)
Now, let's plug in the value x = 16 into the simplified expression:
lim x->16 1/(x) = 1/16
Therefore, the limit of (4 - sqrt(x))/(16x - x^2) as x approaches 16 is equal to 1/16.