Evaluate lim h → 0 [√(9 + h) - 3] / h.

We will use the concepts of limit in order to find the required answer.

Answer: The value of the lim h → 0 [√(9 + h) - 3] / h is 1/6

Let us see how we will use the concept of limit in order to find the required answer.

Explanation:

Given: lim h → 0 [√(9 + h) - 3] / h

let's rationalize the numerator by multiply and divide [√(9 + h) - 3] by [√(9 + h) + 3]

lim h → 0 [√(9 + h) - 3] / h = lim h → 0 [√(9 + h) - 3] [√(9 + h) + 3] / h[√(9 + h) + 3]

= lim h → 0 [{√(9 + h)}² - 3²] / h[√(9 + h) + 3] {since a² - b² = (a + b) (a - b)}

= lim h → 0 [9 + h - 9] / h[√(9 + h) + 3]

= lim h → 0 [h] / h[√(9 + h) + 3]

= lim h → 0 1/[√(9 + h) + 3]

= 1/[√(9) + 3] { on substituting h = 0}

= 1/[3 + 3]

= 1/6

Hence, the value of the lim h → 0 [√(9 + h) - 3] / h is 1/6.