Limits at Infinity and Horizontal Asymptotes
Key Questions

Example 1
#lim_{x to infty}{x5x^3}/{2x^3x+7}# by dividing the numerator and the denominator by
#x^3# ,#=lim_{x to infty}{1/x^25}/{21/x^2+7/x^3}={05}/{20+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.

Answer:
Another perspective...
Explanation:
#color(white)()#
As a Real functionTreating
#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 functionTreated 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 nonzero complex value infinitely many times in any arbitrarily small neighbourhood of#oo# . That is called an essential singularity at infinity. 
Questions
Limits

Introduction to Limits

Determining One Sided Limits

Determining When a Limit does not Exist

Determining Limits Algebraically

Infinite Limits and Vertical Asymptotes

Limits at Infinity and Horizontal Asymptotes

Definition of Continuity at a Point

Classifying Topics of Discontinuity (removable vs. nonremovable)

Determining Limits Graphically

Formal Definition of a Limit at a Point

Continuous Functions

Intemediate Value Theorem

Limits for The Squeeze Theorem