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

Textbook Question

Chapter 2, 2.3 - Problem 41 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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kspcr111 | In Training Educator

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`lim_(x->3) (2x + |x - 2|)`

sol:

`lim_(x->3) (2x + |x - 2|)`

=>`lim_(x->3) (2x )+lim_(x->3) (|x - 2|)`

=> `6+lim_(x->3) (|x - 2|) ` ------------------(1)

as when `x-> 3` ,`|x-2|` is positive so `|x-2|= x-2`

so ,

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

= `lim_(x->3) (x - 2)`

 as `x-> 3`

=> `lim_(x->3) (x - 2) = 3-2 = 1`

so now from (1) we get

`6+lim_(x->3) (|x - 2|) = 6+1 = 7`

so,

`lim_(x->3) (2x + |x - 2|)= 7`

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