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Applications of One-Sided Limits

Determine if a limit exists and, if so, its value.

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Introduction to Limits

Definition of a Limit

Definition:

The notation   limxx0f(x)=L    means that as x approaches (or gets very close to) x, the limit of the function f ( x ) gets very close to the value L

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Basically, a limit is the value a function approaches at a certain point. Limits can be found by:

  • plugging the x-value into the equation
  • looking at a graph and estimating the y-value for a function at that point
  • plugging the equation into a calculator and using a table to see what value the function approaches from the left and right sides
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Write using limit notation:

  1. Write the limit of 7x3+2x+3x5 as x approaches a from the left.
  2. Write the limit of f(m) as m approaches a.
  3. Write the limit of g(z) as z approaches b.

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Find the following limits at x = 0:

1.

x -0.2 -0.1 -0.01 0 0.01 0.1 0.2
) 2.993347 2.998334 2.999983 Undefined 2.999983 2.998334

2.993347

2. 

x -0.2 -0.1 -0.01 0 0.01 0.1 0.2
) 0.993347 0.998334 0.999983 Undefined 1.000001 1.000012 1.000027

3.

 

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Find the following limits:

  1. limx05x2
  2. limx4x
  3. limx0sinxx 
  4. limx03xx+11 
  5. limx01cosxx2
Click here for the answers.

One-Sided Limits

If the value that the function approaches differs on the left and the right, you can use one-sided limits to determine the value. 

What is the limit of this function as x approaches 0 from the left? From the right?

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Limits from the left are written with a - after the number, from the right has a +.

Tip: The sign corresponds to the sides of the y-axis. The right side is positive, the left is negative.


Does the Limit Exist?

For a limit to exist, the limit from the right side must be equal to the limit from the left. If the right-hand limit does not equal the limit from the left then the limit does not existFor example, in the graph above, limx0limx0+. Therefore  limx0 does not exist.

To determine if the limit of a piecewise function (a function with two or more parts) exists, you must see if the right-hand and left-hand limits are equal. 


Remember that we are not concerned about finding the value of ) at but rather near . So, for < 1 (limit from the left),

limx1f(x)=limx1(3x)=(31)=2

and for > 1 (limit from the right),

limx1+f(x)=limx1+(3xx2)=2

Now since the limit exists and is the same on both sides, it follows that

limx1f(x)=2


Practice

Find the following limits:

  1. limx3

  2. limx2+

  3. limx1+ and limx1

  4. limx1

  5. limx2 and limx5+

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Find the following limits based on the equation:

Hint: Graph the equations or look at a table.

  1. limx2+x22x+8x2=
  2. limx0+x2+4xx=
  3. limx1+4x2x3x1=
  4. limx0+x24xx=
  5. limx24x27x2x2=
  6. limx53x213x+10x+5=

Click here for more help.

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Determine if the limits exist:

  1. g(x)={3;x1x+4;x<1
  2. h(x)={2;x15x+2;x<1
  3. g(x)={2;x=23x+3;x2
  4. g(x)={3x4;x=32x1;x3
  5. f(x)={3;x=12;x1

Click here for guidance.

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