Let f(x)= lim as h->0 of ((x+h)^2-x^2)/ h. For what value of x does f(x)= 4?
(x+h)^2-x^2 = x^2 + 2 x h + h^2 - x^2
or in other words 2 x h + h^2
or h (2 x + h)
divide that by h
2x+h
limit as h ---> 0 is
2 x
so
f(x) = 2 x as h--->0
4 = 2 x
x = 2
Well, if we calculate the limit as h approaches 0 for the function f(x), we end up with (2x + h) + 2x. To find the value of x where f(x) equals 4, we can set the equation equal to 4: (2x + h) + 2x = 4. Hmm, this seems like a job for algebra! Let's solve for x by simplifying: 4x + h = 4. Well, if we subtract h from both sides, we get 4x = 4 - h. Now, to isolate x, divide both sides by 4. So, x = 1 - h/4. But remember, h is approaching 0. So, our answer is x = 1. Now, wasn't that a mathematical circus act?
To find the value of x for which f(x) is equal to 4, we substitute the given function f(x) into the equation and solve for x.
Let's start by evaluating the function f(x):
f(x) = lim as h->0 of ((x+h)^2 - x^2) / h
Expanding the numerator:
f(x) = lim as h->0 of (x^2 + 2xh + h^2 - x^2) / h
Canceling out the x^2 terms:
f(x) = lim as h->0 of (2xh + h^2) / h
Factoring out h from the numerator:
f(x) = lim as h->0 of h(2x + h) / h
Canceling out h:
f(x) = lim as h->0 of 2x + h
Taking the limit as h approaches 0, h does not affect the value, so we can replace h with 0:
f(x) = 2x + 0
Simplifying the equation:
f(x) = 2x
Now we set f(x) equal to 4 and solve for x:
2x = 4
Dividing both sides by 2:
x = 2
Therefore, for f(x) to be equal to 4, x must be equal to 2.
To find the value of x for which f(x) equals 4, we need to substitute f(x) = 4 into the equation and solve for x.
Let's start by substituting f(x) with the given expression:
4 = lim as h->0 of ((x+h)^2 - x^2) / h
Now, let's simplify the expression inside the limit:
4 = lim as h->0 of (x^2 + 2xh + h^2 - x^2) / h
The x^2 terms on the numerator cancel out:
4 = lim as h->0 of (2xh + h^2) / h
Now, let's factor out an 'h' from the numerator:
4 = lim as h->0 of h(2x + h) / h
Cancel out the 'h' terms:
4 = lim as h->0 of 2x + h
Since h is approaching 0, as the limit states, we can substitute h with 0 in the equation:
4 = 2x + 0
Simplifying further:
4 = 2x
Divide both sides by 2:
2 = x
Therefore, the value of x for which f(x) is equal to 4 is x = 2.