Introductory Calculus: Limit of a Function and Continuity


external image f17a4cb4-b048-490f-b1ac-b03d26e7693a.gif The limit of a function from left or right or both:

    • If external image 8b2dc7c3-f3f4-44e1-9060-eb3e37d0a9b9.gif, we call the limit of this function as x approaches 1 from the left as
      • external image f0dab082-964a-4b58-a575-5b2f4ff42561.gif

Similarly, we call the limit as x approaches 1 from the right equal to 2.

      • external image 0459ba13-2167-456f-893a-c91f315bb9fb.gif

    • What is mysterious about this concept of limit?

The function obviously has a value of 2 when x = 1.

      • external image 54ad3dd7-ca6b-4eff-ac37-f5cca7dc1102.gif
We say that

      • external image 4593a4ec-88e4-40df-9a7b-a4a3fb960cfd.gif

This is a simple thing to analyze because this function is continuous (the graph is “connected”) and we can clearly see that thefunction value at x = 1 is 2. See the graph below.

external image 6071886a-caf1-4f24-a422-2cf2c6d351d3.gif external image 320cf9e4-890a-449b-8086-33ffda4d47ec.gif

      • Suppose that

      • external image 9f09c463-4736-443d-84bc-bbca6d683e47.gif

    • What is the limit as x approaches –1 from the left/right?
    • We will get two different values for this limit because there are two different expressions for external image 41a0c41d-caf8-4372-90f9-89c0cbca6b15.gif.
      • If x approaches –1 from the left [for external image 9fe19e62-e3f1-499b-80b7-7413d9e307f8.gif ], we use the formula

        • external image db442fb8-6307-480b-8b75-7b1e4106939b.gif

      • which gives us

        • external image b1c47656-85c7-4ded-a594-fcceae9e56e1.gif
      • If x approaches –1 from the right [for external image 29691a9f-4e2e-41f5-a780-beb24aa2636f.gif], we use the formula

        • external image 74830b09-d69b-42e7-a8ff-688a108c01f6.gif

      • which gives us

        • external image b6fbd59a-d2ff-48b2-8534-66ee0f38ba0c.gif
    • In this case, we say that

      • external image 0d240f04-4558-42aa-80fe-8b13d0ecc54c.gif

    • Since these limits are different, we say that the ONE limit as x approaches –1 does not exist. In order for this limit to exist, both the left hand and right hand limits would have to be the same, and the graph would have to “connect” from the left and right sides.

    • external image c6c15cbd-38d4-444d-b25e-c5ca09261776.gif

    • These limits from the left and right have different values. Looking at a graph from a calculator screen, we can see that the left handgraph and the right hand graph do not meet in one point, but the limits from the left and right sides can be seen on the graph as the yvalues of this function for each piecewise-defined part of the graph. Notice the “open” circle and "closed" circle on the graph.
    • Notice that the value of the function external image 3cd448d6-11f0-4ada-85a2-f78a98d61a62.gif at the point –1 is external image aa60321e-b868-4c54-8cb7-9a323970e161.gif because only external image 057b0a48-3d80-4555-914d-3f294a9e28fe.gifdefines this function for the value x = –1. This is an important fact as we examine the continuity of a function. We will compare this value, if it exists, to the limit value.

external image d3c15915-8e46-4865-9a7f-7551ae6254bb.gif Continuity of a function at a point x = a:
Definition
A function is said to be continuous at a point where x = a if three conditions hold:
      • external image 5993961c-5ca1-4a9c-b0ca-d595ca275e75.gif


    • Using our function

      • external image 550110fb-4042-4ccd-a7ca-5326002aee46.gif

      we apply the definition of continuity at the point x = –1.
      • Condition (i) is OK because

        • external image cf4f68c3-2d14-4d3b-8021-bef73da848ef.gifmeaning that external image 12b2c8dd-08d6-44c7-b1e3-955c130a0d40.gif exists.
      • Condition (ii) is not OK because

        • external image 5212d3e5-47ca-4d35-be02-3dfe36bea3b1.gifmeaning external image e345c6b8-1f7a-4338-b270-f58fa66d9c97.gifdoes not exist.
      • We can STOP here because as soon as one of these three conditions goes wrong, we know the function is NOT continuous at the given value of x.


Find the following limits if they exist.

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Examples
Examples
Examples:
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  • Use the following two functions to answer the next set of questions.

    • external image 858940c7-d94a-4bb2-a51b-38206fc25b26.gif and external image 8352c4fd-4090-4c16-b546-356dbba6ef1d.gif


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Examples
Examples
Examples:
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Discuss the continuity of f(x) and g(x) at x = 2.
  • Test your understanding about limits by answering the following two questions about two new functions: h(x) and r(x).


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Examples
Examples
Examples:
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Let external image 5d310cd0-83e1-435c-ab27-015496e33c5a.gif

Graph and discuss continuity at x = 3.

  • external image fb0f6680-9250-48a5-8abc-4e08d3827043.gif

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Let external image b7457ac6-630c-4e63-bb0e-5a87f0447bf0.gif

Graph and discuss continuity at x = 3.

  • external image a696b04d-e9e3-497b-b964-8db1f6231ef3.gif

M Ransom

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