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