Vertical asymptotes & one sided limits

[mikexz]

I noticed in my textbook that they define a vertical asymptote when the limits of one side of the vertical asymptote approaches infinitely the other side must approach negative infinity. Why is that? Why can't the limits of both left and right side be infinity or negative infinity?

thanks

 

[Deleted member 4993]

I noticed in my textbook that they define a vertical asymptote when the limits of one side of the vertical asymptote approaches infinitely the other side must approach negative infinity. Why is that? Why can't the limits of both left and right side be infinity or negative infinity?

thanks

it could be. try

\(\displaystyle f(x)\, = \, \frac{1}{(x\, - \, 1)^2}\)

 

[mikexz]

thanks, it's sort of awkward that the textbook is wrong

I just have one more question, when I doing the second derivative of a function to see whether it's concave up or down, why does that work? I know that when I take the 1st derivative it's the slope of the function but what exactly is the second derivative of the function? My book says it has to do with the tangentsbeing above the function or below, how is that relevent?

thanks again

 

[Deleted member 4993]

thanks, it's sort of awkward that the textbook is wrong

Textbook is not wrong - it just did not give you enough examples.

I just have one more question, when I doing the second derivative of a function to see whether it's concave up or down, why does that work? I know that when I take the 1st derivative it's the slope of the function but what exactly is the second derivative of the function? My book says it has to do with the tangentsbeing above the function or below, how is that relevent? <<< draw convex and concave curves and tangents to those curves - relevancy will be self evident

thanks again

 

[mikexz]

I am just stuck on the idea of a slope of a slope. why can the slope of a slope tell you about the concavity about a function?

 

[Deleted member 4993]

I am just stuck on the idea of a slope of a slope. why can the slope of a slope tell you about the concavity about a function?<<< draw convex and concave curves and tangents to those curves - observe what happens to the magnitude of the slope around the curvature.

around a convex curve - the slope may start from being positive - will go to zero at the max point - then it will go to negative. So what happens to the slope of slope around a maxima (convex)?