lim h->0 ((1/(x+h)^2)-(1/x^2))/h
To simplify the expression, let's first find a common denominator:
((1/(x+h)^2)-(1/x^2))/h
= ((x^2 - (x+h)^2)/(x^2(x+h)^2))/h
= ((x^2 - (x^2 + 2hx + h^2))/(x^2(x+h)^2))/h
= ((-2hx - h^2)/(x^2(x+h)^2))/h
Now, let's cancel out the h term in the numerator and denominator:
= (-2x - h)/(x^2(x+h)^2)
Finally, let's take the limit as h approaches 0:
lim h->0 ((1/(x+h)^2)-(1/x^2))/h
= lim h->0 (-2x - h)/(x^2(x+h)^2)
= -2x/(x^2(x+0)^2)
= -2x/x^2
= -2/x
Therefore, the limit of the expression as h approaches 0 is -2/x.
To find the limit of the given expression as h approaches 0, we can simplify it by combining the fractions:
lim h->0 ((1/(x+h)^2) - (1/x^2))/h
Let's start by simplifying the first fraction:
1/(x+h)^2 = 1/((x+h)*(x+h)) = 1/(x^2 + 2hx + h^2)
Now, let's subtract the second fraction:
(1/(x+h)^2) - (1/x^2) = 1/(x^2 + 2hx + h^2) - 1/x^2
To combine the fractions, we need a common denominator, which is x^2 * (x^2 + 2hx + h^2):
1/(x^2 + 2hx + h^2) - 1/x^2 = (1*x^2 - (x^2 + 2hx + h^2))/(x^2*(x^2 + 2hx + h^2))
Simplifying the numerator:
(1*x^2 - (x^2 + 2hx + h^2))/(x^2*(x^2 + 2hx + h^2)) = (x^2 - x^2 - 2hx - h^2)/(x^2*(x^2 + 2hx + h^2))
= (-2hx - h^2)/(x^2*(x^2 + 2hx + h^2))
Now, let's divide the entire expression by h:
(-2hx - h^2)/(h*x^2*(x^2 + 2hx + h^2))
Canceling out one h in the numerator and denominator:
(-2x - h)/(x^2*(x^2 + 2hx + h^2))
Finally, let's evaluate the limit as h approaches 0:
lim h->0 (-2x - h)/(x^2*(x^2 + 2hx + h^2))
Now, substitute h = 0 into the expression:
(-2x - 0)/(x^2*(x^2 + 2(0)x + 0^2))
= -2x/(x^2*(x^2))
Simplifying further:
-2x/(x^2*(x^2)) = -2/(x^2)
Therefore, the limit of the given expression as h approaches 0 is -2/(x^2).
To evaluate the given limit, let's simplify the expression step by step:
Step 1: Simplify the numerator
((1/(x+h)^2)-(1/x^2))
To simplify, we need to find a common denominator. The common denominator for these fractions is x^2(x+h)^2.
The expression becomes:
[(x^2-(x+h)^2)/(x^2(x+h)^2)]
Step 2: Expand and simplify the numerator
Expanding the numerator (x^2-(x+h)^2):
= (x^2 - (x^2 + 2xh + h^2))
= (x^2 - x^2 - 2xh - h^2)
= -2xh - h^2
The expression becomes:
[(-2xh - h^2)/(x^2(x+h)^2)]
Step 3: Divide by h
Now, divide the expression by h:
[(-2xh - h^2)/(h * x^2(x+h)^2)]
Step 4: Cancel out common terms
We can cancel out an h from the numerator and the denominator:
[(-2x - h)/(x^2(x+h)^2)]
Step 5: Take the limit
Now, substitute h with 0 in the expression since we are trying to find the limit as x approaches 0:
[(-2x - 0)/(x^2(x+0)^2)]
= (-2x)/(x^2 * 0^2)
= (-2x)/(0)
= undefined
Therefore, the limit does not exist for the given expression as x approaches 0.