Limits With Absolute Value, Help!?

Question

Find the limit of x as it approaches -5 from left and right.

(x+5) / |x+5|

Answer 1

As x approaches -5 from right, limit is 1

As x approaches -5 from left, limit is -1

Reason is as so:

The numerator is x + 5

The denominator is |x+5|

When going to -5 from the right, the x + 5 is a positive number y

The numerator is y

The denominator is |y| = y since we have determined its positive

y/y = 1

When going to -5 from the left, the x + 5 is a negative number g

The numerator is -g [I wrote a negative in front of the g to clarify that it is a negative number]

The denominator is |-g| = g

g/-g = 1

Note that with each example, I replaced x + 5 with some variable (y and g) that I know will either be positive or negative.

Hope this helps.

I hope this helps

Answer 2

As x nears - 5 from the left, the numerator is always negative, while the denominator is always positive; so the limit is - 1 as x approaches

- 5 from the left.

On the other hand, as x nears - 5 from the right (i.e., x > - 5), the numerator is always positive, so the limit is + 1 as x approaches

- 5 from the right.  

Answer 3

-1 & + 1....|x+5| = -(x+5) if x ≤ -5

.........................= (x+5) if  -5 ≤ x