Evaluate each limit if it exists. Show all work and give evidence of reasoning. If a limit does not exist, identify in what way it does not exist (∞,−∞, DNE). limx→1|x−1|x2−1

Question

Evaluate each limit if it exists. Show all work and give evidence of reasoning. If a limit does not exist, identify in what way it does not exist (∞,−∞, DNE).  limx→1|x−1|x2−1

lamusesamuset · Accepted Answer

limx→1(|x−1|x2−1)  If limx→a−f(x)≠limx→a+f(x) then the limit does not exist  limx→1+(|x−1|x2−1)  x-1 is positive when x→1+. Therefore |x−1|=x−1  =limx→1+(x−1x2−1)  Simplify x−1x2−1:1x+1  x−1x2−1  Factor x2−1:(x+1)(x−1)  x2−1  Rewrite 1 as 12  =x2−12  Apply Difference of Two Squares Formula: x2−y2=(x+y)(x−y)  x2−12=(x+1)(x−1)  =(x+1)(x−1)  =x−1(x+1)(x−1)  Cancel the common factor: x-1  =1x+1  =limx→1+(1x+1)  Plug in the value x=1  =11+1  Simplify  =12  limx→1+(|x−1|x2−1)=12  x−1 is negative when x→1−. Therefore |x−1|=−x+1  =limx→1−(−x+1x2−1)  Simplify −x+1x2−1: −1x+1

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Evaluate each limit if it exists. Show all work and give evidence of reasoning.

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Answer & Explanation

New Questions in Calculus 1