The discussion revolves around evaluating the limits of the function (x-4)/(x-2) as x approaches 2 from both the right (2^+) and the left (2^-). The user initially misinterprets the behavior of the function near the vertical asymptote at x = 2, mistakenly concluding that the limit approaches +∞ from the right. However, after clarification, it is established that as x approaches 2 from the right, (x-4) approaches -2, leading to the correct conclusion that the limit is -∞. The user also references Wolfram Alpha for verification, which confirms the correct limit values.
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murmillo said:drawing a picture will help, i think.
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.
I did not because I want to follow the method that my instructor usedMark44 said:Did you follow murmillo's suggestion?
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?
And what method is that?Nitrate said:I did not because I want to follow the method that my instructor used
i did however draw the asymptote
SammyS said:And what method is that?
For \displaystyle \lim_{x\to2^+} \frac{x-4}{x-2}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)
Homework Equations
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..
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 !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.
SammyS said:DUH !
I made a mistake !
... , as x approaches 2 from the right, x-4 is close to -2 (negative 2).
Therefore, this gives you (-2)/(0+) → -∞
The whole statement is, when x is close to 2, x - 4 is close to -2.Nitrate said:sorry, but I'm having trouble understanding why x-4 is close to -2
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.Nitrate said:I did not because I want to follow the method that my instructor used
i did however draw the asymptote