pre calc

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find all the horizontal and vertical asymptoes of the functions

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

q(x)=(x-1)/x

  • pre calc -

    When the denominator equals zero, there is a vertical asymptoe

    When the numerator becomes constant as x approaches very large, there is a horizontal asymptoe.

    For instance, in the second

    q(x)=(x-1)/x= 1-1/x as x gets very large, it becomes 1-0 or 1

    In the first,
    f(x)=x/(x-1)= 1/(1-1/x) (multiplied numerator and denominator by 1/x)
    f(x)= 1/(1-0) when x is very large, or f(x)=1

  • pre calc -

    x = 1 for the first one

    x = 0 for the second

  • pre calc -

    yes, for the vertical asy.

  • pre calc -

    so what about this one g(x)=1.5^x

  • pre calc -

    No vertical, no horizontal.

  • pre calc -

    how can you tell it has neither

  • pre calc -

    As Mr. Bob mentioned:
    When the denominator becomes zero when x=c and c is finite, there is a vertical asymptote.
    A horizontal asymptote is typically identified by the fact that lim f(x) approaches a constant value as x->∞ or -> -&infin.
    In the case of:
    g(x)=1.5^x
    there is no denominator that makes g(x) infinite when x is finite, so no vertical asymptote.
    g(x) becomes infinite when x->+∞, so no horizontal asymptote on the right. But on the left..., as x->-∞, g(x) approaches zero, so what do you think?

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