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:

    #color(white)()#
    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#?

    #color(white)()#
    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