limit h->0 [sqrt((4+h)) - 2] / h = ?
Thanks
multiply numerator and denominator by [sqrt(4+h)+2]
numerator becomes h
denominator becomes h([sqrt(4+h)+2])
the h/h becomes 1, now take the limit.
To compute the limit of the expression (sqrt(4+h) - 2) / h as h approaches 0, you can follow these steps:
Step 1: Multiply the numerator and denominator by the conjugate of the numerator, which is (sqrt(4+h) + 2). This is done to eliminate the square root in the numerator.
(sqrt(4+h) - 2) / h * (sqrt(4+h) + 2) / (sqrt(4+h) + 2)
Step 2: Simplify the numerator by using the difference of squares formula.
[(4+h) - 2^2] / (h * (sqrt(4+h) + 2))
[(4+h) - 4] / (h * (sqrt(4+h) + 2))
h / (h * (sqrt(4+h) + 2))
Step 3: Simplify the expression by canceling out the common factor of h in the numerator and denominator.
1 / (sqrt(4+h) + 2)
Step 4: Take the limit as h approaches 0.
lim(h->0) 1 / (sqrt(4+h) + 2)
Plugging in h = 0 into the expression gives:
1 / (sqrt(4+0) + 2) = 1 / (2 + 2) = 1 / 4
Therefore, the limit of the expression (sqrt(4+h) - 2) / h as h approaches 0 is 1/4.