Find the limit
Limit as h approaches 0 of :
SqRt(4+h)-2
____________
h
by relating it to the derivative. (Indicate reasoning.)
L'Hopital's rule works, is one way. That is most practial way to solve this.
To find the limit as h approaches 0, of the expression (Sqrt(4+h) - 2) / h, we can relate it to the derivative of a function and apply a fundamental concept in calculus called the definition of the derivative.
Let's begin by representing the given expression as a function. Let f(x) = Sqrt(4+x) - 2. We want to find the limit of f(x) as x approaches 0.
The derivative of f(x) with respect to x, denoted by f'(x) or df(x)/dx, represents the instantaneous rate of change of the function at a specific point. It indicates how the function behaves as x changes infinitesimally.
Using the definition of the derivative, we have:
f'(x) = lim(h->0) (f(x+h) - f(x)) / h
Let's apply this definition to our function f(x):
f'(x) = lim(h->0) (Sqrt(4 + x + h) - 2 - (Sqrt(4 + x) - 2)) / h
By simplifying the expression, we obtain:
f'(x) = lim(h->0) (Sqrt(4 + x + h) - Sqrt(4 + x)) / h
Notice that this is similar to the expression we want to evaluate the limit for. By comparing the two expressions, we can see that our limit is equal to the derivative of f(x) evaluated at x = 0:
lim(h->0) (Sqrt(4 + h) - 2) / h = f'(0)
Thus, to find the limit, we need to calculate the derivative of f(x) at x = 0 and evaluate it.
Let's differentiate f(x) using the power rule and chain rule:
f'(x) = (1/2)(4 + x)^(-1/2)
Now, we can evaluate the derivative at x = 0:
f'(0) = (1/2)(4 + 0)^(-1/2) = (1/2)(4)^(-1/2) = (1/2)(1/2) = 1/4
Therefore, the limit as h approaches 0 of (Sqrt(4 + h) - 2) / h is equal to 1/4.
In summary, by relating the given expression to the derivative of the function and using the definition of the derivative, we found that the limit is equal to the derivative of the function evaluated at x = 0, which resulted in 1/4.