CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 14. Concepts of Calculus

1. Megan is being extremely liberal with the idea of “=∞” because what she really means for the two limits is
   “DNE”. For the function evaluated at 2 the correct response is “undefined”. Two things that do not exist cannot be
   equal to one another.


2. 
   

a. xlim→ 1 −( 2 x− 1 ) = 2 · 1 − 1 = 2 − 1 = 1

b.x→−lim 3 +(x+^22 )=− 32 + 2 =−^21 =− 2

c. limx→ 2 +

(x (^3) − 8
x− 2

)

=xlim→ 2 +

((x− 2 )(x (^2) + 2 x+ 4 )
(x− 2 )

)

=xlim→ 2 +(x^2 + 2 x+ 4 ) = 22 + 2 · 2 + 4 = 12

3. Use the definition of continuity.
   
   
   • limx→ 1 −f(x) = (− 1 )^2 − 1 = 1 − 1 = 0
   
   
   • f(− 1 ) = 3
   
   
   • x→−lim 1 +f(x) =− 1 + 3 = 2
   
   
   
   

xlim→a−f(x)^6 =f(a)^6 =xlim→a+f(x)so this function is discontinuous atx=−1. It is continuous everywhere else.

Practice

Evaluate the following limits.

1. limx→ 6 −( 3 x^2 − 4 )


2. limx→ 0 −^3 x−x^1


3. limx→ 0 +^3 x−x^1


4. limx→ 0 +|xx|


5. limx→ 0 −|xx|


6. limx→ 0 +

√x

√ 1 +√x− 1

Consider

f(x) =







2 x^2 − 1 x< 1

1 x= 1

−x+ 2 1 <x

7. What is limx→ 1 −f(x)?


8. What is limx→ 1 +f(x)?


9. Isf(x)continuous atx=1?
   Consider

g(x) =







4 x^2 + 2 x− 1 x<− 2

8 x=− 2

− 3 x+ 5 − 2 <x