Limits With Absolute Value!! Please Help!!?

Question

Find the limit of x as it approaches 4 from the left and right.

|x-4| / (x-4)

Please explain, I really don't understand.

Answer 1

ok here is what you do, as X approaches from the right, meaning the right of the number line, so try plug in numbers like 7 then 6 then 5, and see what it's approaching.  Do the same thing with the left pick numbers like 1, 2, 3 plug them in and see what it's approaching, if the number that it's becoming from approaching the two sides is the same, ( i know this sentence is worded terribly but bear with me)  then the limit is valid and hence that is your answer, i believe is one, although check for yourself.

Answer 2

f(x) = |x-4| / (x-4)

(a) When x-->4 from the right, x> 4, |x - 4| =(x - 4)

f(x) = (x - 4) / (x - 4)

lim f(x) = lim (x - 4 )/(x - 4) = 1

x-->4+ . .x-->4+

(b) When x-->4 from the left, x < 4, |x - 4| =- (x - 4)

f(x) = - (x - 4) / (x - 4)

lim f(x) = lim -(x - 4 )/(x - 4) = -1

x-->4- . .x-->4-

Answer 3

As x nears 4 uniformly "from the left" (i.e., as x is slighly less than 4), the given expression equals - 1, since the numerator is always positive while the denominator is always negative. Limit = - 1.

On the other hand, as x nears 4 uniformly "from the right", the ratio is always positive and therefore the limit for that case = +1.

Answer 4

The equation can be simplified.

When x>4, the equation would be (x-4)/(x-4) = 1

This is the limit from the right.

When x< 4, the equation would be -(x-4)/(x-4) = -1

This is the limit from the left

Another method would be to graph it.

Answer 5

if x>4

then |x-4|=(x-4)

and |x-4|/(x-4)=(x-4)/(x-4)=1

if x<4

then |x-4|=-(x-4)

and |x-4|/(x-4)=-(x-4)/(x-4)=-1

if you have problems with this questions

you can instance 3 or 4 numbers near that number

for example in this question 4.0001 & -4.0001

Answer 6

(absolute value of x-4) divided by (x-4)

you would have to know the value of x to solve the problem.