Approaching Positive or Negative Infinity?

Jason76

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The problem below has a limit that DNE. How do we find out whether it approaches positive or negative infinity? Hint. :confused:

 
The limit goes toward \displaystyle \infty, but how do we know that? :confused:
 
How do we know it doesn't have asymptotes at points of discontinuity? :confused:
 
How do we know it doesn't have asymptotes at points of discontinuity? :confused:
We know that because the left and right limits are both real numbers.

limx0(x21)=01=1.\displaystyle \displaystyle \lim_{x \rightarrow 0^-}(x^2 - 1) = 0 - 1 = - 1. The function is approaching a point, not a line.

limx0+(2x+1)=2+1=3.\displaystyle \displaystyle \lim_{x \rightarrow 0^+}(2x + 1) = 2 + 1 = 3. Again the function is approaching a point, not a line.

The reason that this function is not continuous at x = 0 is because it has no limit as x approaches 0, and the reason for that is not because the right and left limits do not exist; they do exist, but one of them is not equal to f(0).

Does that make sense?
 
This is the problem I'm really trying to ask, and Iv'e figured it out.

Assuming that you got a DNE for a limit and it's asymptotic, then in order to know whether it goes to positive or negative infinity, without looking at the graph, you do this:

Look at the quotient of the final working of the equation.

If at the end you got:

n\displaystyle n = some number

z\displaystyle z = some number

limxz=[n0]=\displaystyle \lim x \rightarrow z = [\dfrac{n}{0-}] = -\infty

limxz=[n0]=\displaystyle \lim x \rightarrow z = [\dfrac{-n}{0-}] = \infty

limxz=[n0+]=\displaystyle \lim x \rightarrow z = [\dfrac{n}{0+}] = \infty

limxz=[n0+]=\displaystyle \lim x \rightarrow z = [\dfrac{-n}{0+}] = -\infty
 
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This is the problem I'm really trying to ask, and Iv'e figured it out.

Assuming that you got a DNE for a limit and it's asymptotic, then in order to know whether it goes to positive or negative infinity, without looking at the graph, you do this:

Look at the quotient of the final working of the equation.

If at the end you got:

n\displaystyle n = some number

z\displaystyle z = some number

limxz=[n0]=\displaystyle \lim x \rightarrow z = [\dfrac{n}{0-}] = -\infty

limxz=[n0]=\displaystyle \lim x \rightarrow z = [\dfrac{-n}{0-}] = \infty

limxz=[n0+]=\displaystyle \lim x \rightarrow z = [\dfrac{n}{0+}] = \infty

limxz=[n0+]=\displaystyle \lim x \rightarrow z = [\dfrac{-n}{0+}] = -\infty
I am not sure that I grasp what you are saying. It seems a bit fuzzy to me.

For example,

f(x)=1x    limx0f(x)= and limx0+f(x)=+.\displaystyle \displaystyle f(x) = \dfrac{1}{x} \implies \lim_{x \rightarrow 0^-}f(x) = - \infty\ and\ \lim_{x \rightarrow 0^+}f(x) = + \infty.

So what can you say about the limit as x approaches zero? Nothing as far as I am concerned. You can't say it is plus infinity or negative infinity. You can only say that it does not exist because the right and left limits are not equal.
 
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