# How do you use a graph to show that the limit does not exist?

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

(Left-Hand Limit#ne# Right-Hand 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 violent oscillation

- there is a vertical asymptote

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To show that the limit does not exist using a graph, you need to demonstrate that the function does not approach a single value as the input approaches a certain point. This can be done by identifying at least two different paths or approaches that yield different limit values or by showing that the function oscillates or has a jump at the specific point.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the limit of #(2x-8)/(sqrt(x) -2)# as x approaches 4?
- How do you find the limit of # ( x+sin 2x)/( 3x) # as x approaches 0?
- For all #x>=0# and #4x-9<=f(x)<=x^2-4x+7# how do you find the limit of f(x) as #x->4?
- How do you evaluate #(3e^-x+6)/(6e^-x+3)# as x approaches infinity?
- How do you find the limit of #x^(2x)# as x approaches 0?

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