Question

question 15 evaluate the limits. if a limit does not exist, enter
\\(\lim_{x\to -1^+} \frac{|x + 1|}{x + 1} =\\)
\\(\lim_{x\to -1^-} \frac{|x + 1|}{x + 1} =\\)
\\(\lim_{x\to -1} \frac{|x + 1|}{x + 1} =\\)

Explanation:

Step1: Right limit: define absolute value

For $x \to -1^+$, $x > -1$, so $x+1 > 0$. Thus $|x+1|=x+1$.
$\lim_{x \to -1^+} \frac{|x+1|}{x+1} = \lim_{x \to -1^+} \frac{x+1}{x+1}$

Step2: Simplify right limit expression

Cancel $x+1$ (non-zero here):
$\lim_{x \to -1^+} 1 = 1$

Step3: Left limit: define absolute value

For $x \to -1^-$, $x < -1$, so $x+1 < 0$. Thus $|x+1|=-(x+1)$.
$\lim_{x \to -1^-} \frac{|x+1|}{x+1} = \lim_{x \to -1^-} \frac{-(x+1)}{x+1}$

Step4: Simplify left limit expression

Cancel $x+1$ (non-zero here):
$\lim_{x \to -1^-} -1 = -1$

Step5: Check overall limit existence

Right limit $
eq$ left limit, so overall limit does not exist.

Answer:

$\lim_{x \to -1^+} \frac{|x+1|}{x+1} = 1$
$\lim_{x \to -1^-} \frac{|x+1|}{x+1} = -1$
$\lim_{x \to -1} \frac{|x+1|}{x+1}$ does not exist