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.

Explain Bot Explain Bot answered
8 months ago

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.

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