Solve lim x->7+ as....sqrt(x-7)^3/(x-7). I found that it equals 0, but I don't know if I did everything right.
To solve the limit, we can simplify the expression first. Let's start by simplifying the numerator, sqrt(x-7)^3.
The cube of a square root can be rewritten as the square root of the cube. Therefore, sqrt(x-7)^3 can be rewritten as (sqrt(x-7))^2 * sqrt(x-7).
Now, let's simplify the denominator, (x-7).
Since we are taking the limit as x approaches 7 from the right (x -> 7+), we can see that the denominator (x-7) approaches 0 as x approaches 7 from the right.
Now, let's put the simplified numerator and denominator together:
[(sqrt(x-7))^2 * sqrt(x-7)] / (x-7)
The next step is to evaluate the limit expression. As we said earlier, we know that the denominator approaches 0 as x approaches 7 from the right (x -> 7+).
To evaluate the limit, we can substitute the value of x into the expression and see what we get:
[(sqrt(x-7))^2 * sqrt(x-7)] / (x-7)
[(sqrt(7-7))^2 * sqrt(7-7)] / (7-7)
[(sqrt(0))^2 * sqrt(0)] / 0
[0 * 0] / 0
0 / 0
At this point, we have encountered an indeterminate form, where both the numerator and denominator are equal to zero.
Therefore, we cannot determine the limit value just by evaluating it directly. We need to try a different method to find the limit.
One way to proceed is to use L'Hôpital's rule. L'Hôpital's rule states that if we have an indeterminate form of 0/0 or ∞/∞, we can differentiate the numerator and denominator separately and then take the limit again.
Let's differentiate the numerator and denominator separately and then evaluate the limit again:
Taking the derivative of the numerator:
d/dx [(sqrt(x-7))^2 * sqrt(x-7)] = 2 * sqrt(x-7) * (1/2) * (1/sqrt(x-7)) * sqrt(x-7) + sqrt(x-7) * (3/2) * (1/sqrt(x-7))^2
= 1 * sqrt(x-7) + (3/2) * (sqrt(x-7))^2
= sqrt(x-7) + (3/2) * (x-7)
Taking the derivative of the denominator:
d/dx (x-7) = 1
Now, let's evaluate the limit again using the derivatives:
lim x->7+ [(sqrt(x-7))^3 / (x-7)]
= lim x->7+ [sqrt(x-7) + (3/2) * (x-7)] / 1
= [sqrt(7-7) + (3/2) * (7-7)] / 1
= [sqrt(0) + (3/2) * 0] / 1
= [0 + 0] / 1
= 0 / 1
= 0
So, after evaluating the limit using L'Hôpital's rule, we find that the limit of the given expression as x approaches 7 from the right (x->7+) is 0.
Therefore, your answer is correct. The limit of the expression is indeed 0.