Determine whether the limit exists: L :

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Here are the solutions for problems 8, 9, and 10.

8. Determine whether the limit exists: $\lim_{x \to 0} \frac{|x|}{x}$

Evaluate the right-hand limit.
For $x > 0$ , $|x| = x$ .
$\lim_{x \to 0^+} \frac{|x|}{x} = \lim_{x \to 0^+} \frac{x}{x} = \lim_{x \to 0^+} 1 = 1$
Evaluate the left-hand limit.
For $x < 0$ , $|x| = -x$ .
$\lim_{x \to 0^-} \frac{|x|}{x} = \lim_{x \to 0^-} \frac{-x}{x} = \lim_{x \to 0^-} (-1) = -1$
Compare the one-sided limits.
Since $\lim_{x \to 0^+} \frac{|x|}{x} \neq \lim_{x \to 0^-} \frac{|x|}{x}$ ( $1 \neq -1$ ), the limit does not exist.
The limit is $\boxed{\text{Does not exist}}$ .

9. Determine whether the limit exists: $\lim_{x \to 2} \frac{|x - 2|}{x - 2}$
Evaluate the right-hand limit.
For $x > 2$ , $x - 2 > 0$ , so $|x - 2| = x - 2$ .
$\lim_{x \to 2^+} \frac{|x - 2|}{x - 2} = \lim_{x \to 2^+} \frac{x - 2}{x - 2} = \lim_{x \to 2^+} 1 = 1$
Evaluate the left-hand limit.
For $x < 2$ , $x - 2 < 0$ , so $|x - 2| = -(x - 2)$ .
$\lim_{x \to 2^-} \frac{|x - 2|}{x - 2} = \lim_{x \to 2^-} \frac{-(x - 2)}{x - 2} = \lim_{x \to 2^-} (-1) = -1$
Compare the one-sided limits.
Since $\lim_{x \to 2^+} \frac{|x - 2|}{x - 2} \neq \lim_{x \to 2^-} \frac{|x - 2|}{x - 2}$ ( $1 \neq -1$ ), the limit does not exist.
The limit is $\boxed{\text{Does not exist}}$ .

10. Determine whether the limit exists: $\lim_{x \to 0} \frac{x}{|x|}$
Evaluate the right-hand limit.
For $x > 0$ , $|x| = x$ .
$\lim_{x \to 0^+} \frac{x}{|x|} = \lim_{x \to 0^+} \frac{x}{x} = \lim_{x \to 0^+} 1 = 1$
Evaluate the left-hand limit.
For $x < 0$ , $|x| = -x$ .
$\lim_{x \to 0^-} \frac{x}{|x|} = \lim_{x \to 0^-} \frac{x}{-x} = \lim_{x \to 0^-} (-1) = -1$
Compare the one-sided limits.
Since $\lim_{x \to 0^+} \frac{x}{|x|} \neq \lim_{x \to 0^-} \frac{x}{|x|}$ ( $1 \neq -1$ ), the limit does not exist.
The limit is $\boxed{\text{Does not exist}}$ .

Determine whether the limit exists: L :

Step-by-step solution

Evaluate the right-hand limit.

Evaluate the left-hand limit.

Compare the one-sided limits.

Evaluate the right-hand limit.

Evaluate the left-hand limit.

Compare the one-sided limits.

Evaluate the right-hand limit.

Evaluate the left-hand limit.

Compare the one-sided limits.

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