explain why the limit does not exist. \nlim_{x\\to0}\\frac{x}{|x|}\nselect the correct choice below and fill…

explain why the limit does not exist. \nlim_{x\\to0}\\frac{x}{|x|}\nselect the correct choice below and fill in the answer boxes within your choice.\na. as x approaches 0 from the left, \\frac{x}{|x|} approaches \\square. as x approaches 0 from the right, \\frac{x}{|x|} approaches \\square. this means there is no single number l that the\nfunction values get arbitrarily close to as x\\to0.\nb. the function is not defined at x = 0, so the limit cannot exist.

explain why the limit does not exist. \nlim_{x\\to0}\\frac{x}{|x|}\nselect the correct choice below and fill in the answer boxes within your choice.\na. as x approaches 0 from the left, \\frac{x}{|x|} approaches \\square. as x approaches 0 from the right, \\frac{x}{|x|} approaches \\square. this means there is no single number l that the\nfunction values get arbitrarily close to as x\\to0.\nb. the function is not defined at x = 0, so the limit cannot exist.

Answer

Explanation:

Step1: Recall the definition of absolute value

For (x<0), (|x|=-x). So when (x\to0^{-}) (approaching (0) from the left), (\frac{x}{|x|}=\frac{x}{-x}=- 1).

Step2: Recall the definition of absolute value for positive (x)

For (x > 0), (|x|=x). So when (x\to0^{+}) (approaching (0) from the right), (\frac{x}{|x|}=\frac{x}{x}=1).

Answer:

A. As (x) approaches (0) from the left, (\frac{x}{|x|}) approaches (-1). As (x) approaches (0) from the right, (\frac{x}{|x|}) approaches (1). This means there is no single number (L) that the function values get arbitrarily close to as (x\to0).