use numerical or graphical evidence to determine the left and right hand limits of the $lim_{x\rightarrow4^{…

use numerical or graphical evidence to determine the left and right hand limits of the $lim_{x\rightarrow4^{-}}\frac{|4x - 16|}{4 - x}=$ $lim_{x\rightarrow4^{+}}\frac{|4x - 16|}{4 - x}=$ question help: message instructor

use numerical or graphical evidence to determine the left and right hand limits of the $lim_{x\rightarrow4^{-}}\frac{|4x - 16|}{4 - x}=$ $lim_{x\rightarrow4^{+}}\frac{|4x - 16|}{4 - x}=$ question help: message instructor

Answer

Explanation:

Step1: Analyze the absolute - value function

First, rewrite (|4x - 16|=|4(x - 4)| = 4|x - 4|).

Step2: Calculate the left - hand limit ((x\to4^{-}))

When (x\to4^{-}), (x-4\lt0), so (|x - 4|=-(x - 4)). Then (\lim_{x\to4^{-}}\frac{|4x - 16|}{4 - x}=\lim_{x\to4^{-}}\frac{4|x - 4|}{4 - x}=\lim_{x\to4^{-}}\frac{4-(x - 4)}{4 - x}). Simplify the expression: (\lim_{x\to4^{-}}\frac{-4(x - 4)}{4 - x}=\lim_{x\to4^{-}}\frac{-4(x - 4)}{-(x - 4)} = 4).

Step3: Calculate the right - hand limit ((x\to4^{+}))

When (x\to4^{+}), (x - 4\gt0), so (|x - 4|=x - 4). Then (\lim_{x\to4^{+}}\frac{|4x - 16|}{4 - x}=\lim_{x\to4^{+}}\frac{4|x - 4|}{4 - x}=\lim_{x\to4^{+}}\frac{4(x - 4)}{4 - x}). Since (x-4) and (4 - x) are negatives of each other, (\lim_{x\to4^{+}}\frac{4(x - 4)}{4 - x}=-4).

Answer:

(\lim_{x\to4^{-}}\frac{|4x - 16|}{4 - x}=4) (\lim_{x\to4^{+}}\frac{|4x - 16|}{4 - x}=-4)