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

Answer 1

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:

  1. there is a jump discontinuity
    (Left-Hand Limit #ne# Right-Hand Limit)
    The limit does not exist at #x=1# in the graph below.

    1. 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.

      1. there is a violent oscillation
        (e.g., #sin(1/x)# at #x=0#, shown below)

        I hope that this was helpful.

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Answer 2

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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Answer from HIX Tutor

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.

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.

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