Evaluate the limit of the function [(3+h)^2-9]/h, if h approaches to 0.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the value of lim h-->0[[(3+h)^2-9]/h]

lim h-->0[[(3+h)^2-9]/h]

=> lim h-->0[[(3 + h - 3)(3 + h + 3)/(3 + h - 3)]

cancel (3 + h - 3)

=> lim h-->0[[(3 + h + 3)]

substitute h = 0

=> 6

The required value of lim h-->0[[(3+h)^2-9]/h] = 6

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

If we'll let h=o, we'll get f(0) undefined.

Since h is cancelling the numerator, that means that h is the root of the numerator.

We'll expand the square from numerator and we'll get:

(3+h)^2 = 9 + 6h + h^2

We'll subtract 9 both sides:

(3+h)^2 - 9 = 6h + h^2

We'll re-write the limit of the function:

lim (6h + h^2)/h = lim h(6+h)/h

lim h(6+h)/h = lim (6+h) = 6 + 0 = 6

The limit of the function, if h approaches to 0, is lim [(3+h)^2-9]/h = 6.

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