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Limits: Graphical Solutions
Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:
The limits are defined as the value that the function approaches as it goes to an x value. Using this definition, it is possible to find the value of the limits given a graph. A few examples are below:
In general, you can see that these limits are equal to the value of the function. This is true if the function is continuous. Continuity Continuity of a graph is loosely defined as the ability to draw a graph without having to lift your pencil. To better understand this, see the graph below:
Let’s investigate at the flowing points:
Discontinuous at this point The value is not defined at - 3 “Removable discontinuity”
Discontinuous at this point The limit of the left is not equal to the limit from the right “Jump discontinuity”
Discontinuous at this point The limit from the left is equal to the right, but is not equal to the value of the function “Removable discontinuity”
Continuous at this point The limit from the left is equal to the limit from the right and equal to the value of the function
Continuous at this point The limit from the left is equal to the limit from the right and equal to the value of the function
Discontinuous at this point The value of the limit is equal to negative infinity and therefore not defined “Infinite discontinuity”
Limits
One-Sided Limits: General Definition One-sided limits are differentiated as right-hand limits (when the limit approaches from the right) and left-hand limits (when the limit approaches from the left) whereas ordinary limits are sometimes referred to as two-sided limits. Right-hand limits approach the specified point from positive infinity. Left- hand limits approach this point from negative infinity. The right-handed limit:
The left-handed limit:
A. Now you try some!
Determine if the following limits exists:
A More Formal Definition of Continuity
From this information, a more formal definition can be found. Continuity, at a point a , is defined when the limit of the function from the left equals the limit from the right and this value is also equal to the value of the function. Using notation, for all points a where , the function is said to be continuous.
3. (^ )
B. Now you try some!
Limits of Rational Functions: Substitution Method A rational function is a function that can be written as the ratio of two algebraic expressions. If a function is considered rational and the denominator is not zero, the limit can be found by substitution. This can be seen in the example below (which is similar to the example #3 above, but now done in one quick, convenient step):
This can be defined more formally as: If and are algebraic expressions and , then:
C. Now you try some!
Factoring Method
Consider the function. How would you find the limit of as approaches -3? If you try to
use substitution to find the limit, world-ending paradoxes ensue:
But fear not, this answer just tells us that we must use a different method to find the limit, because the function likely has a “hole” at the given x value. Therefore, the factoring method can be tried. To start this method, the numerator and denominator must be factored (in this case the denominator is “factored” already).
The factor can be canceled to get the much simpler limit expression of that can easily be evaluated via substitution:
Therefore, the result of the limit can be found, with the understanding that there is a “hole” in the graph
at. Hence,.
D. Now you try some! a) b)
Conjugate Method
The conjugate of a binomial expression (i.e. an expression with two terms, you can tell this because of the Latin root bi- meaning two ) is the same expression with opposite middle signs. For example, the conjugate of (^) √ is (^) √. This is really useful if you have a radical in your limit. This is because the product of two conjugates containing radicals will, itself, contain no radical expressions. See below:
(√ )(√ ) √^ √ √
You should use the conjugate method whenever you have a limit problem containing radicals for which substitution does not work.
Example:
Evaluate √
First try the substitution method:
√ √
Well, another hole in the universe, or at least the graph. Indicating that you’ll need another method to find the limit since the function probably has a hole at. To start, multiply both the numerator and denominator by the conjugate of the radical expression (√ ):
√ (√ ) (√ )
So, if |^ |^ |^ |^ |^ |^ ( ) therefore, | | | | and.
Limits: Advanced Topics
Previously, when we found that the result of a limit doing straight substitution yielded we used
factoring or conjugation to be able to solve the problem. What happens when neither of those methods prove useful? You become very grateful for the 17th-century French mathematician Guillaume de L’Hôpital. L’Hôpital was the man that derived a method of solving these types of equations, known as indeterminate forms. This method, known as L’Hôpital’s Rule, is formally defined below.
Example 1: indeterminate form of
Find the limit
Using L’Hôpital’s Rule:
Example 2: indeterminate form of
Find the limit
Using L’Hôpital’s Rule:
Using L’Hôpital’s Rule again:
Formal Definition: L’Hôpital’s Rule
If the limit results in one of the following forms:
And exits and , then:
Example 3: indeterminate form of
Find the limit ( )
( ) Using L’Hôpital’s Rule: ( )
Example 4: indeterminate form of
Find the limit
Let. Then Using L’Hôpital’s Rule:
Therefore
Example 5: indeterminate form of Find the limit
Using L’Hôpital’s Rule:
F. Now you try some!
Why does the √^ not equal (^) ⁄ (^) √?
Note the trick that is needed here – what happens if you use L’Hôpital’s Rule without making this initial change?
Examples: a.
whereas
b. [Note that is meaningless since x! is not defined for negative values.] c. = + (Note that as well [why?])
If the degrees are equal, then is equal to the leading coefficient of n(x) over the
leading coefficient of d(x).
Examples: a.
b. =
⁄ ⁄
G. Now you try some!
Solutions to “Now You Try Some!”
A.
DNE (“Does not exist”)
B.
- 2c+
- 1/
C.
- 5/
- 5/
D.
- 13
- 3
E.
- 4
- 1/
F.
- 3/
- 3/
- (gets larger and larger without bound; technically this limit DNE)
- DNE – cannot take the logarithm of negative values
- 1
- Upon substituting -1 into the numerator and denominator, we do not get one of the indeterminate forms; therefore we cannot use L’Hôpital’s Rule. More advanced methods need to be used.
G.
- 2/
- 0
- 0
- +
- +
