<meta http-equiv="refresh" content="1; url=/nojavascript/"> Applications of One-Sided Limits ( Study Aids ) | Analysis | CK-12 Foundation
Dismiss
Skip Navigation

Applications of One-Sided Limits

%
Best Score
Practice
Best Score
%
Practice Now
Introduction to Limits
 0  0  0

Feel free to modify and personalize this study guide by clicking “Customize.”

Definition of a Limit

Definition:

The notation   \lim_{x \rightarrow x_0} f(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

.

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
.

Write using limit notation:

  1. Write the limit of 7x^3 + \sqrt{2x} + 3x - 5 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  .

.

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.

 

.

Find the following limits:

  1. \lim_{x \rightarrow 0} \frac{5x} {2}
  2. \lim_{x \rightarrow 4} \sqrt{x}
  3. \lim_{x \rightarrow 0} \frac{sin x} {x} 
  4. \lim_{x \rightarrow 0} \frac{3x} {\sqrt{x + 1} - 1} 
  5. \lim_{x \rightarrow 0} \frac{1-cos x} {x^2}
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?

.

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, \lim_{x\to0^-} \ne \lim_{x\to0^+}. Therefore  \lim_{x\to0} 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),

\lim_{x \rightarrow 1^{-}} f(x) = \lim_{x \rightarrow 1^{-}} (3 - x) = (3 - 1) = 2

and for > 1 (limit from the right),

\lim_{x \rightarrow 1^+} f(x) = \lim_{x \rightarrow 1^+} (3x - x^2) = 2

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

\lim_{x \rightarrow 1} f(x) = 2


Practice

Find the following limits:

  1. \lim_{x\to-3^-}

  2. \lim_{x\to2^+}

  3. \lim_{x\to-1^+} and \lim_{x\to-1^-}

  4. \lim_{x\to-1}

  5. \lim_{x\to-2^-} and \lim_{x\to5^+}

.

Find the following limits based on the equation:

Hint: Graph the equations or look at a table.

  1. \lim_{x\to2^+}\frac{-x^2 - 2x + 8}{x - 2}=
  2. \lim_{x\to0^+}\frac{-x^2 + 4x}{x}=
  3. \lim_{x\to1^+}\frac{4x^2 - x - 3}{x - 1}=
  4. \lim_{x\to0^+}\frac{x^2 - 4x}{x}=
  5. \lim_{x\to2^-}\frac{4x^2 - 7x - 2}{x - 2}=
  6. \lim_{x \to -5^-}\frac{-3x^2 - 13x + 10}{x + 5}=

Click here for more help.

.

Determine if the limits exist:

  1.   g(x)= \begin{cases}  3 ; x \geq -1\\  x + 4 ; x < -1\\  \end{cases}
  2.   h(x)= \begin{cases}  -2; x \geq -1\\  -5x + 2 ; x < -1\\  \end{cases}
  3.  g(x)= \begin{cases} -2 ; x = - 2\\ -3x + 3 ; x \not= -2\\ \end{cases}
  4.   g(x)= \begin{cases}  -3x - 4 ; x = 3\\  -2x - 1 ; x \not= 3\\  \end{cases}
  5.   f(x)= \begin{cases}  -3 ; x = -1\\  -2 ; x \not= -1\\  \end{cases}

Click here for guidance.

Image Attributions

Reviews

Email Verified
Well done! You've successfully verified the email address .
OK
Please wait...
Please wait...

Original text