# Having trouble evaluating these limits

• Nitrate
In summary, the limit as x approaches 2 from the right of (x-4)/(x-2) is -∞ because as x gets closer to 2, x-4 gets closer to -2, resulting in a negative value in the numerator and a very small positive value in the denominator, which gives a large negative result. This was confirmed by drawing a graph of the function.

## Homework Statement

The limit as x approaches 2^+ of (x-4)/(x-2)

The limit as x approaches 2^- of (x-4)/(x-2)

## The Attempt at a Solution

Figured out that the domain is x ≠ 2, therefore 2 is a vertical asymptote
not sure where to input values/what to do really..

drawing a picture will help, i think.

murmillo said:
drawing a picture will help, i think.

I tried
lim
x→2+ (0+-4)/(0+-2)
but i keep getting + infinity

and I'm not even sure if that's how you approach this question.

Can anyone help?

Last edited:
murmillo said:
drawing a picture will help, i think.

Nitrate said:
I tried
lim
x→2+ (0+-4)/(0+-2)
but i keep getting + infinity

and I'm not even sure if that's how you approach this question.

As x approaches 2 from the right, what does the graph of y = (x - 4)/(x - 2) do?
As x approaches 2 from the left, what does the graph of y = (x - 4)/(x - 2) do?

Mark44 said:

As x approaches 2 from the right, what does the graph of y = (x - 4)/(x - 2) do?
As x approaches 2 from the left, what does the graph of y = (x - 4)/(x - 2) do?
I did not because I want to follow the method that my instructor used
i did however draw the asymptote

Nitrate said:
I did not because I want to follow the method that my instructor used
i did however draw the asymptote
And what method is that?

SammyS said:
And what method is that?

That's what I'm trying to piece together.
there was something about 1. (+)/(0^-) = - infinity
2. (-)/(0^-1) = + infinity
3. (-2)/(0^+)= - infinity
4. (1)/(0^+) = infinity

it has worked so far for my previous examples such as:
lim
x→1+ (3/x-1)

Domain: x can not equal 1
V.A: x = 1

lim (3)/(0+-1) = + infinity

but it is not working for the example that I'm on and i don't know why

Nitrate said:

## Homework Statement

The limit as x approaches 2^+ of (x-4)/(x-2)

The limit as x approaches 2^- of (x-4)/(x-2)

## The Attempt at a Solution

Figured out that the domain is x ≠ 2, therefore 2 is a vertical asymptote
not sure where to input values/what to do really..
For $\displaystyle \lim_{x\to2^+} \frac{x-4}{x-2}$
As x approaches 2 from the right (from the positive x direction) x-2 → 0+, that is to say x-2 becomes very small, but is positive.

Also, as x approaches 2 from the right, x-4 is close to positive 2.

So this gives you (+2)/(0+) → +∞​

The limit as x → 2- can be handled similarly.

SammyS said:
For $\displaystyle \lim_{x\to2^+} \frac{x-4}{x-2}$
As x approaches 2 from the right (from the positive x direction) x-2 → 0+, that is to say x-2 becomes very small, but is positive.

Also, as x approaches 2 from the right, x-4 is close to positive 2.

So this gives you (+2)/(0+) → +∞​

The limit as x → 2- can be handled similarly.

however, for the first limit, the answer in the back of the book comes up as -∞
and when i type the limit as x approaches 2^+ of (x-4)/(x-2) into wolfram alpha it also comes up as - infinity
http://www.wolframalpha.com/input/?i=the+limit+as+x+approaches+2^++of+(x-4)/(x-2)

can anyone else help?

SammyS said:
For $\displaystyle \lim_{x\to2^+} \frac{x-4}{x-2}$
As x approaches 2 from the right (from the positive x direction) x-2 → 0+, that is to say x-2 becomes very small, but is positive.

Also, as x approaches 2 from the right, x-4 is close to positive 2.

So this gives you (+2)/(0+) → +∞​

The limit as x → 2- can be handled similarly.
DUH !

... , as x approaches 2 from the right, x-4 is close to -2 (negative 2).

Therefore, this gives you (-2)/(0+) → -∞

SammyS said:
DUH !

... , as x approaches 2 from the right, x-4 is close to -2 (negative 2).

Therefore, this gives you (-2)/(0+) → -∞

sorry, but I'm having trouble understanding why x-4 is close to -2

Nitrate said:
sorry, but I'm having trouble understanding why x-4 is close to -2
The whole statement is, when x is close to 2, x - 4 is close to -2.

Nitrate said:
I did not because I want to follow the method that my instructor used
i did however draw the asymptote
You should still sketch a graph of the function. The intent wasn't meant to be a proof, but to enable you to have some insight as to what the limit should be.

finally got the concept!
thanks everyone :)