`lim_(x->0.5^-) (2x + 12)/(|2x^3 - x^2|)` Find the limit, if it exists. If the limit does not exist, explain why.

Textbook Question

Chapter 2, 2.3 - Problem 43 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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kspcr111 | In Training Educator

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`lim_(x->0.5^-) (2x + 12)/(|2x^3 - x^2|)`

sol:

`lim_(x->0.5^-) (2x + 12)/(|2x^3 - x^2|)`

=>`(lim_(x->0.5^-) (2x + 12))/(lim_(x->0.5^-) (|2x^3 - x^2|))` ------(1)

in the numerator ,we get

`(lim_(x->0.5^-) (2x + 12))`

= `2(0.5) + 12 = 1 +12 = 13`

in the denominator we get

`(lim_(x->0.5^-) (|2x^3 - x^2|))`

as when `x-> 0.5^- ` so `|2x^3 - x^2|` is negatiive

so, 

`|2x^3 - x^2| = -(2x^3 - x^2)= x^2 - 2x^3`

so, `(lim_(x->0.5^-) (|2x^3 - x^2|)) =(lim_(x->0.5^-) (x^2 - 2x^3))`

when approaching to 0 the denominator is a positive quantity so,

`(lim_(x->0.5^-) (x^2 - 2x^3)) = 0^+`

Now, from (1)

`(lim_(x->0.5^-) (2x + 12))/(lim_(x->0.5^-) (|2x^3 - x^2|)) = 13/0^+ = + oo`

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