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Finding Limits: Properties of Limits |
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Consider the rational function{: data-type="term" .no-emphasis}
The function can be factored as follows:
Does this mean the function f
is the same as the function g(x)=x+1?
The answer is no. Function f
does not have x=7
in its domain, but g
does. Graphically, we observe there is a hole in the graph of f( x )
at x=7,
as shown in [link] and no such hole in the graph of g( x ),
as shown in [link].
{: #CNX_Precalc_Figure_12_02_001}
{: #CNX_Precalc_Figure_12_02_002}
So, do these two different functions also have different limits as x
approaches 7?
Not necessarily. Remember, in determining a limit{: data-type="term" .no-emphasis} of a function as x
approaches a,
what matters is whether the output approaches a real number as we get close to x=a.
The existence of a limit does not depend on what happens when x
equals a.
Look again at [link] and [link]. Notice that in both graphs, as x
approaches 7, the output values approach 8. This means
Remember that when determining a limit, the concern is what occurs near x=a,
not at x=a.
In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.
Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits{: data-type="term"}, which is a collection of theorems for finding limits.
Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.
and B
represent real numbers, and f
and g
be functions, such that lim x→a f(x)=A
and lim x→a g(x)=B.
For limits that exist and are finite, the properties of limits are summarized in [link]
| Constant, k | lim x→a k=k |
| Constant times a function | lim x→a [ k⋅f(x) ]=k lim x→a f(x)=kA |
| Sum of functions | lim x→a [ f(x)+g(x) ]= lim x→a f(x)+ lim x→a g(x)=A+B |
| Difference of functions | lim x→a [ f(x)−g(x) ]= lim x→a f(x)− lim x→a g(x)=A−B |
| Product of functions | lim x→a [ f(x)⋅g(x) ]= lim x→a f(x)⋅ lim x→a g(x)=A⋅B |
| Quotient of functions | lim x→a f(x) g(x) = lim x→a f(x) lim x→a g(x) = A B ,B≠0 |
| Function raised to an exponent | lim x→a [f(x)] n = [ lim x→∞ f(x) ] n = A n , where n is a positive integer |
| nth root of a function, where n is a positive integer | lim x→a f(x) n = lim x→a [ f(x) ] n = A n |
| Polynomial function | lim x→a p(x)=p(a) |
Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit{: data-type="term" .no-emphasis} of a polynomial function as x
approaches a
is equivalent to simply evaluating the function for a
.
-
Use the properties of limits to break up the polynomial into individual terms.
-
Find the limits of the individual terms.
-
Add the limits together.
-
Alternatively, evaluate the function for a
. {: type="1"}
When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit{: data-type="term" .no-emphasis} of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.
approaches 2 and raise the result to the 5th power.
, can we still determine the limit of the function as x
approaches a
?**
Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.
Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.
- Factor the numerator and denominator completely.
- Simplify by dividing any factors common to the numerator and denominator.
- Evaluate the resulting limit, remembering to use the correct domain. {: type="1"}
Notice, the function
-
If the quotient as given is not in indeterminate ( 0 0 )
form, evaluate directly.
-
Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the least common denominator (LCD){: data-type="term" .no-emphasis}.
-
If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the conjugate{: data-type="term" .no-emphasis} of the numerator. Recall that a± b
are conjugates.
-
Simplify.
-
Evaluate the resulting limit. {: type="1"}
- Try factoring or finding the LCD.
- If the limit{: data-type="term"} cannot be found, choose several values close to and on either side of the input where the function is undefined.
- Use the numeric evidence to estimate the limits on both sides. {: type="1"}
so we will try values close to 7 from the left and the right.
Left-hand limit: | 6.9−7 | 6.9−7 = | 6.99−7 | 6.99−7 = | 6.999−7 | 6.999−7 =−1
Right-hand limit: | 7.1−7 | 7.1−7 = | 7.01−7 | 7.01−7 = | 7.001−7 | 7.001−7 =1
Since the left- and right-hand limits are not equal, there is no limit.
- Determine a Limit Analytically {: data-display="block"}
- The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See [link].
- The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See [link] and [link].
- The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See [link].
- The limit of the root of a function equals the corresponding root of the limit of the function.
- One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See [link].
- Another method of finding the limit of a complex fraction is to find the LCD. See [link].
- A limit containing a function containing a root may be evaluated using a conjugate. See [link].
- The limits of some functions expressed as quotients can be found by factoring. See [link].
- One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See [link].
whose limit, as x
approaches a,
is f( a ).
is a polynomial function, the limit of a polynomial function as x
approaches a
will always be f( a ).
approaches a
and the result is f( a )= 0 0 ,
does this mean that the limit of f
does not exist?
as x
approaches c ,
is undefined?
approaches c,
or (2) the left and right-hand limits are not equal.
For the following exercises, evaluate the limits algebraically.
For the following exercise, use the given information to evaluate the limits: lim x→c f(x)=3,
lim x→c g( x )=5
For the following exercises, evaluate the following limits.
For the following exercises, find the average rate of change f(x+h)−f(x) h .
{: #CNX_Precalc_Figure_12_02_201}
{: #CNX_Precalc_Figure_12_02_202}
For the following exercises, refer to [link].
{: #CNX_Precalc_Figure_12_02_203}
approaches 0?
approaches 0?
gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval [ 1,2 ]
.
Find the average rate of change of the height from t=1
second to t=1.5
seconds.
years compounded continuously at 4.25% interest is given by the formula A= A 0 e 0.0425t ,
where A 0
is the initial amount invested. Find the average rate of change of the balance of the account from t=1
year to t=2
years if the initial amount invested is $1,000.00.
properties of limits : a collection of theorems for finding limits of functions by performing mathematical operations on the limits