How do you use the formal definition of a limit to prove #lim(x^3 + 3x + 5) = 19# as x approaches 2?

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Answer 1 · 2015-10-31T18:00:29

See the explanation section below.

Sketch of preliminary analysis

#x^3+3x+5-19 = x^3+3x-14=(x-2)(x^2+2x+7)#

#abs(x-2) < 1# implies #abs(x^2+2x+7) < 22#

Proof

Given #epsilon > 0#, choose #delta = min{1,epsilon/22}#

Now if #0 < abs(x-2) < delta#, then #abs(x^2+2x+7) < 22#, and

#abs((x^3+3x+5)-19) = abs(x-2) abs(x^2+2x+7)#

# < delta * 22 <= epsilon/22 * 22 = epsilon#

That is: #abs((x^3+3x+5)-19) < epsilon#