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Provided by the Academic Center for Excellence 1 Calculus Limits November 2013
Calculus Limits
Images in this handout were obtained from the My Math Lab Briggs online e-book.
A limit is the value a function approaches as the input value gets closer to a specified quantity. Limits are used to define continuity, derivatives, and integrals. This handout focuses on determining limits analytically and determining limits by looking at a graph.
Determining a Limit Analytically There are many methods to determine a limit analytically, and they are usually used in succession. First, see if the limit can be evaluated by direct substitution. Second, if direct substitution yields an undefined result, factor and reduce the fraction or multiply by the conjugate. Third, if the second method does not work, find the left and right sided limits.
- Evaluate by Direct Substitution: Take the value of the limit and evaluate the function at this value. There are two situations where direct substitution will be used, direct substitution with a numerical value and direct substitution with infinity.
Example 1: Limits at a Numerical Value.
lim ๐ฅ๐ฅโ
โ 2 ๐ฅ๐ฅ^3 + 9 + 3๐ฅ๐ฅ โ 1
๏ฟฝ2(2)^3 + 9 + 3(2) โ 1
โ^25 + 5
Example 2: Limits at Infinity. Limits at infinity are solved by plugging in โ or โโ into the function for the given variable. When determining limits at infinity, think more about the trends of the function at infinity rather than the math.
lim ๐ฅ๐ฅโโโ
๐ฅ๐ฅ^2
(โโ)^2
Think about the decimal value of a fraction with a large number in the denominator. What is the trend? As the denominator gets larger, the fraction as a whole gets smaller until it ultimately reaches zero. Evaluating using arbitrary large numbers for infinity will show this trend:
10 100
Therefore, it can be said that a constant numerator divided by infinity is equal to zero.
2 + 0 = 2
lim ๐ฅ๐ฅโโโ
๐ฅ๐ฅ^2
Some Common Trends at Infinity
๐๐ โ
a(โ)๐๐ b(โ)๐๐^
(โโ)๐๐^ = โโ
(ยฑโ)๐๐^ = โ
(ยฑโ)โ๐๐^ = 0
- Factor and Reduce or Multiply by the Conjugate: Use when the first method yields an undefined solution.
Example 1 : Factoring and reducing.
lim ๐ฅ๐ฅโ
โ๐ฅ๐ฅ^ โ^2
If ๐ฅ๐ฅ were simply evaluated at 4 as shown in the first method, it would yield a zero in the denominator; therefore, the slope is undefined. One way to avoid this is to factor the numerator and denominator if applicable.
Example 1: Piecewise functions.
To find the limit as ๐ฅ๐ฅ approaches 1 from the left side, the first equation must be used because it defines the function at values less than and equal to one. Since the equation will not yield an undefined result, direct substitution can be used.
lim ๐ฅ๐ฅโ1โ
To find the limit as ๐ฅ๐ฅ approaches 1 from the right side, the second equation must be used because it defines the function at values greater than one. Since the equation will not yield an undefined result, direct substitution can be used.
lim ๐ฅ๐ฅโ1+^
The limit of this function does not exist (DNE) because the values for the left and right sided limits as ๐ฅ๐ฅ approaches 1 yields two different answers.
lim ๐ฅ๐ฅโ
Example 2: Infinite Limits. Sometimes when computing limits, an answer of โ or โโ will be reached, resulting in an infinite limit.
lim ๐ฅ๐ฅโ
(๐ฅ๐ฅ โ 1)^2 (๐ฅ๐ฅ โ 3)
Direct substitution will yield a denominator of zero and the function is already reduced to its simplest form so left and right sided limits must be used. The graph of this function is shown on the next page in Figure 2.27.
Figure 2.27 obtained from: http://media.pearsoncmg.com/aw/aw_briggs_calculus_et_1/ebook/bccalcet01_0204.nbp
To evaluate the left and right sided limits, evaluate the function for values very close to the limit. Think about the decimal value of a fraction with a small number in the denominator. What is the trend? As the denominator gets smaller the fraction as a whole gets larger until it ultimately reaches infinity. Evaluating using numbers close to the limit will show this trend:
To evaluate the limit as ๐ฅ๐ฅ approached one from the right, try evaluating with a number slightly greater than one.
lim ๐ฅ๐ฅโ1+^
[(1.01) โ 1]^2 [(1.01) โ 3]
Now, try evaluating with a number closer to 1.
lim ๐ฅ๐ฅโ1+^
[(1.000001)^ โ 1]^2 [(1.000001) โ 3]
To evaluate the limit as ๐ฅ๐ฅ approaches one from the left, try evaluating with a number slightly less than 1.
lim ๐ฅ๐ฅโ1โ^
[(0.9)^ โ 1]^2 [(0.9)^ โ 3]
Now, try evaluating with a number closer to 1.
lim ๐ฅ๐ฅโ1โ^
[(0.999999)^ โ 1]^2 [(0.999999)^ โ 3]
Looking at the graph as ๐ฅ๐ฅ approaches 1 from the right side, the function approaches 3.
lim ๐ฅ๐ฅโ1+^
Since both the left-sided and right-sided limits have the same value, a limit exists for this function. lim ๐ฅ๐ฅโ
Example 2 : Sometimes a discontinuity will not result in a limit. Using Figure 2.45 again, look at the limit as ๐ฅ๐ฅ approaches 3. As ๐ฅ๐ฅ approaches 3 from the left side, the function approaches 2. lim ๐ฅ๐ฅโ3โ^
As ๐ฅ๐ฅ approaches 3 from the right side, the function (y value) approaches 1.
lim ๐ฅ๐ฅโ3+^
The function does not have a limit at ๐ฅ๐ฅ = 3 because the left-sided and right-sided limits have different values at ๐ฅ๐ฅ = 3. lim ๐ฅ๐ฅโ
Example 3: Again, using Figure 2.45, look at the limit as ๐ฅ๐ฅ approaches 5. Since there is an asymptote at ๐ฅ๐ฅ = 5, the graph is discontinuous at ๐ฅ๐ฅ = 5. Left-sided and right-sided limits will have to be used to determine if there is a limit. As ๐ฅ๐ฅ approaches 5 from the left side, the function approaches positive infinity.
lim ๐๐โ5โ^
As ๐ฅ๐ฅ approaches 5 from the left side, the function approaches positive infinity.
lim ๐ฅ๐ฅโ5+^
Since both the left-sided and right-sided limits have the same value, a limit exists for this function. lim ๐ฅ๐ฅโ
