limit of (|x+h|-|x|)/h as h goes to 0 at x=4
why is the answer 1?
observe that we can further simplify (|x+h|-|x|)/h to
(x + h - x)/h
(h)/h
1
rewriting,
lim 1 as h->0 at x = 4 is equal to 1, since the limit of a constant is also equal to that constant.
hope this helps~ :)
thank you, i got it
To find the limit of the given expression, we can simplify it and evaluate it as h approaches 0. Let's proceed step by step:
1. Start with the expression:
(|x+h| - |x|) / h
2. Expand the absolute values using the definition of absolute value:
(|x + h| - |x|) / h = (|x + h - x|) / h = h / h
3. Cancel out the common factor h in the numerator and denominator:
h / h = 1
So, based on simplification, the limit of the expression as h approaches 0 is 1.
Now let's understand why the answer is 1. When h approaches 0, we are essentially looking at the behavior of the function at values infinitely close to x=4.
If we evaluate the expression at x=4 with a small value for h, we find:
(|4+h| - |4|) / h
Substituting h=0, we have:
(|4+0| - |4|) / 0 = (|4| - |4|) / 0
Since |4| is 4 and subtracting |4| from 4 gives us 0, we have:
0 / 0
At this point, we have an indeterminate form since we don't have enough information about the function's behavior exactly at x=4. However, if we apply the limit, we can see that the numerator approaches 0 and the denominator approaches 0 as h approaches 0.
In this scenario, we can apply L'Hôpital's rule, which states that for an indeterminate form 0/0, the limit of the ratio of the derivatives is equal to the limit of the original expression. Deriving both the numerator and denominator, we get:
Derivative of the numerator:
d/dh (|4+h| - |4|) = d/dh (|4+h|) - d/dh (|4|) = 1 - 0 = 1
Derivative of the denominator:
d/dh (h) = 1
Taking the ratio of the derivatives:
(1) / (1) = 1
Thus, by applying L'Hôpital's rule, we can conclude that the limit of the given expression as h approaches 0 at x=4 is indeed 1.