find the limit, if it exists. (if an answer does not exist, enter dne.)\n\\lim_{x\\to3}(9x + |x - 3|)

Answer

Explanation:

Step1: Analyze absolute - value function

When (x\to3), consider the definition of (|x - 3|). When (x\geq3), (|x - 3|=x - 3); when (x<3), (|x - 3|=3 - x). We find the left - hand limit and right - hand limit.

Step2: Calculate left - hand limit ((x\to3^{-}))

For (x<3), (|x - 3|=3 - x). Then (\lim_{x\to3^{-}}(9x+|x - 3|)=\lim_{x\to3^{-}}(9x+(3 - x))=\lim_{x\to3^{-}}(8x + 3)). Substitute (x = 3) into (8x+3), we get (8\times3+3=24 + 3=27).

Step3: Calculate right - hand limit ((x\to3^{+}))

For (x\geq3), (|x - 3|=x - 3). Then (\lim_{x\to3^{+}}(9x+|x - 3|)=\lim_{x\to3^{+}}(9x+(x - 3))=\lim_{x\to3^{+}}(10x-3)). Substitute (x = 3) into (10x - 3), we get (10\times3-3=30 - 3=27).

Step4: Determine the limit

Since (\lim_{x\to3^{-}}(9x+|x - 3|)=\lim_{x\to3^{+}}(9x+|x - 3|)=27), then (\lim_{x\to3}(9x+|x - 3|)=27).