Determining When a Limit does not Exist
Key Questions

Remember that limits represent the tendency of a function, so limits do not exist if we cannot determine the tendency of the function to a single point. Graphically, limits do not exist when:
 there is a jump discontinuity
(LeftHand Limit#ne# RightHand Limit)
The limit does not exist at#x=1# in the graph below.
 there is a vertical asymptote
(Infinit Limit)
(Caution: When you have infinite limits, limits do not exist.)
The limit at#x=2# does not exist in the graph below.
 there is a violent oscillation
(e.g.,#sin(1/x)# at#x=0# , shown below)
I hope that this was helpful.
 there is a jump discontinuity

=
#oo# Solution
#=lim_(x→4)((x4)/(x^28x+16))# , plugging the limit we get#0/0# #=lim_(x→4)((x4)/(x^24x4x+16))# #=lim_(x→4)((x4)/(x(x4)4(x4)))# #=lim_(x→4)(x4)/((x4)^2)# #=lim_(x→4)1/((x4))# Now applying the limit, we get
#=1/0=oo# , which implies limit does not exist 
In short, the limit does not exist if there is a lack of continuity in the neighbourhood about the value of interest.
Recall that there doesn't need to be continuity at the value of interest, just the neighbourhood is required.
Most limits DNE when
#lim_(x>a^)f(x)!=lim_(x>a^+)f(x)# , that is, the leftside limit does not match the rightside limit. This typically occurs in piecewise or step functions (such as round, floor, and ceiling).A common misunderstanding is that limits DNE when there is a point discontinuity in rational functions. On the contrary, the limit exists perfectly at the point of discontinuity!
So, an example of a function that doesn't have any limits anywhere is
#f(x) = {x=1, x in QQ; x=0, otherwise}# . This function is not continuous because we can always find an irrational number between 2 rational numbers and vice versa.
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