6. - / 1 points find the limit, if it exists. (if an answer does not exist, enter dne.) $lim_{x\rightarrow…

6. - / 1 points find the limit, if it exists. (if an answer does not exist, enter dne.) $lim_{x\rightarrow - 5}(|x + 5| - 7x)$ resources read it

Answer

Explanation:

Step1: Analyze the absolute - value function

When (x\to - 5), consider the expression inside the absolute - value (x + 5). When (x\to - 5), for the absolute - value (|x + 5|), when (x\to - 5), (|x + 5|=-(x + 5)) since (x+5\lt0) when (x) is approaching (-5) from the left and right in the neighborhood of (x=-5).

Step2: Substitute the absolute - value expression

Substitute (|x + 5|=-(x + 5)) into the limit (\lim_{x\to - 5}(|x + 5|-7x)), we get (\lim_{x\to - 5}[-(x + 5)-7x]).

Step3: Simplify the expression inside the limit

Expand and simplify (-(x + 5)-7x=-x - 5-7x=-8x - 5).

Step4: Evaluate the limit

Now, find (\lim_{x\to - 5}(-8x - 5)). Using the limit rules (\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)) and (\lim_{x\to a}cx = c\lim_{x\to a}x) ((c) is a constant), we have (\lim_{x\to - 5}(-8x - 5)=-8\lim_{x\to - 5}x-\lim_{x\to - 5}5). Since (\lim_{x\to - 5}x=-5) and (\lim_{x\to - 5}5 = 5), then (-8\times(-5)-5=40 - 5=35).