http://www.ck12.org Chapter 14. Concepts of Calculus
Translate the following mathematical statement into words.
nlim→∞
n
i∑= 1( 1
2)i= 1Solution:The limit of the sum of^12 +^14 +^18 +···as the number of terms approaches infinity is 1.
Example C
Use limit notation to represent the following mathematical statement.
(^13) + (^19) + 271 +···= (^12)
Solution:
nlim→∞
n
i∑= 1
( 1
3)i= 1
2
Concept Problem Revisited
The limit of^8 x^43 +x (^44) +x^36 +x (^23) +x^29 −x^10 asxapproaches infinity is^83. This can be written using limit notation as:
xlim→∞
( 8 x (^4) + 4 x (^3) + 3 x (^2) − 10
3 x^4 + 6 x^2 + 9 x
)
=^83
Vocabulary
Limit notationis a way of expressing the fact that the function gets arbitrarily close to a value. In calculus or analysis
you may define a limit in terms of the Greek letter epsilonεand deltaδ.
Guided Practice
- Describe the end behavior of the following rational function at infinity and negative infinity using limits.
f(x) =− 105 xx (^33) ++ 34 xx (^22) +− 9810
- Translate the following limit expression into words.
hlim→ 0
(f(x+h)−f(x)
h)
=x- What do you notice about the limit expression in # 2?
Answers: - Since the function has equal powers ofxin the numerator and in the denominator, the end behavior is−^12 as
xgoes to both positive and negative infinity.
xlim→∞
(− 5 x (^3) + 4 x (^2) − 10
10 x^3 + 3 x^2 + 98
)
=x→−lim∞(− 5 x (^3) + 4 x (^2) − 10
10 x^3 + 3 x^2 + 98
)
=−^12
- The limit of the ratio of the difference betweenfof quantityxplushandfofxandhashapproaches 0 isx.
- You should notice thath→0 does not meanh=0 because if it did then you could not have a 0 in the denominator.
You should also note that in the numerator,f(x+h)andf(x)are going to be super close together ashapproaches
zero. Calculus will enable you to deal with problems that seem to look like^00 and∞∞.
Practice
Describe the end behavior of the following rational functions at infinity and negative infinity using limits.
1.f(x) =^25 xx^44 ++^43 xx^2 +− 91
2.g(x) =^82 xx^33 ++^44 xx^2 +− 71