Limits at Infinity and Horizontal Asymptotes

Key Questions

  • How do you find limits as x approaches infinity?

    Example 1

    lim_{x to infty}{x-5x^3}/{2x^3-x+7}

    by dividing the numerator and the denominator by x^3,

    =lim_{x to infty}{1/x^2-5}/{2-1/x^2+7/x^3}={0-5}/{2-0+0}=-5/2

    Example 2

    lim_{x to -infty}xe^x

    since -infty cdot 0 is an indeterminate form, by rewriting,

    =lim_{x to -infty}x/e^{-x}

    by l'Hopital's Rule,

    =lim_{x to -infty}1/{-e^{-x}}=1/{-infty}=0

    I hope that this was helpful.

    Wataru · 1 · Nov 7 2014
  • What is the limit as x approaches infinity of e^x?

    Answer:

    Another perspective...

    Explanation:

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    As a Real function

    Treating e^x as a function of Real values of x, it has the following properties:

    • The domain of e^x is the whole of RR.

    • The range of e^x is (0, oo).

    • e^x is continuous on the whole of RR and infinitely differentiable, with d/(dx) e^x = e^x.

    • e^x is one to one, so has a well defined inverse function (ln x) from (0, oo) onto RR.

    • lim_(x->+oo) e^x = +oo

    • lim_(x->-oo) e^x = 0

    At first sight this answers the question, but what about Complex values of x?

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    As a Complex function

    Treated as a function of Complex values of x, e^x has the properties:

    • The domain of e^x is the whole of CC.

    • The range of e^x is CC "\" { 0 }.

    • e^x is continuous on the whole of CC and infinitely differentiable, with d/(dx) e^x = e^x.

    • e^x is many to one, so has no inverse function. The definition of ln x can be extended to a function from CC "\" { 0 } into CC, typically onto { x + iy : x in RR, y in (- pi, pi] }.

    What do we mean by the limit of e^x as x -> "infinity" in this context?

    From the origin, we can head off towards "infinity" in all sorts of ways.

    For example, if we just set off along the imaginary axis, the value of e^x just goes round and around the unit circle.

    If we choose any complex number c = r(cos theta + i sin theta), then following the line ln r + it for t in RR as t->+oo, the value of e^(ln r + it) will take the value c infinitely many times.

    We can project the Complex plane onto a sphere called the Riemann sphere CC_oo, with an additional point called oo. This allows us to picture the "neighbourhood of oo" and think about the behaviour of the function e^x there.

    From our preceding observations, e^x takes every non-zero complex value infinitely many times in any arbitrarily small neighbourhood of oo. That is called an essential singularity at infinity.

    George C. · 12 · Aug 23 2017

Questions

Limits