12.1 Review on Limits and Continuity

Limits (Part 1)

Basic Calculus

Objectives

Limits

The idea of limits is fundamental to the study of calculus. It suggests a boundary that may be reached or not, or possibly exceeded. A mathematical limit has characteristics similar to those of a physical limit. It is the analysis of how function values or outputs change, when inputs change.

Limits

Example

Multiple Choice

What number is the $$\lim_{x\rightarrow2}\left(1+3x\right)$$  approaches from the left?

• 6

• 6.9

• 7

• 7.1

Multiple Choice

What number is the  $$\lim_{x\rightarrow2}\left(1+3x\right)$$  approaches from the right?

• 7

• 7.3

• 7.6

• 8

Multiple Choice

Given that  $$f\left(x\right)=1+3x$$  , what is  $$f\left(2\right)$$  ?

• 6

• 7

• 8

Limits

Multiple Choice

What is the limit of  $$\lim_{x\rightarrow-1}\left(x^2+1\right)$$  ?

• 1

• 2

• 3

• 4

Fill in the Blanks

From the table, determine the number x is approaching.

Fill in the Blanks

From the table, determine the number f(x) is approaching.

Fill in the Blanks

From the table, determine the number x is approaching.

Fill in the Blanks

From the table, determine the number g(x) is approaching.

Multiple Choice

Determine which statement is TRUE based on the given information.

• The limit as x approaches 1 from the left is 2

• The limit as x approaches 1 is 1

• The limit as x approaches 1 does not exist

• There is an infinite discontinuity at x =1

Multiple Choice

f(x) is discontinuous. Determine the type of discontinuity at x = 1

• infinite discontinuity

• point discontinuity

• jump discontinuity

• the function is continuous at x = 1

Multiple Choice

$$\lim_{x\rightarrow-1^-}\ f\left(x\right)=?$$  Use the graph. 

• -2

• 3

• does not exist

• -1

Multiple Choice

$$f\left(x\right)=\frac{1}{x-3}$$  

Graph the function and determine the following:  

$$\lim_{x\rightarrow3}\ f\left(x\right)=?$$  

• does not exist

• 1,000

• -1,000

• 0

Multiple Choice

At which x-value is there a jump discontinuity?

• 3

• 1

• -2

• there is no jump discontinuity

Multiple Choice

At which x-value is there a point discontinuity?

• -2

• 0

• 1

• 3

Multiple Choice

:

• 5

• 1

• -5

• Does not exist

Multiple Choice

What is the limit?

• DNE

• Infinity

• 6

• 12

Multiple Choice

What is the limits of the function when x approching a?

• Undefined

• 2

• Does not exist

Multiple Choice

check image

• True

• False

Multiple Choice

What is the $$\lim_{x\rightarrow-5}\ \frac{x^2-25}{x+5}$$  ?

• d.n.e.

• 5

• 10

• -10

Multiple Choice

check image

• ∞
• DNE
• 3
• 1

Fill in the Blanks

$\lim_{x\rightarrow\infty}$   $\frac{4-x^2}{x^2-1}$  is

Multiple Choice

If $$f\left(x\right)=\frac{8x^3-8}{4x^3-2}$$ , find the value of the following limit.

• 2

• 4

• 8

• +∞

Multiple Choice

What is this limit?

• $$\infty$$

• $$-\infty$$

• 0

• -2/3

• DNE

Multiple Choice

Find the limit. Hint: Where is the horizontal asymptote?

• ^-2/₉

• 0

• $$\infty$$

• $$-\infty$$

Multiple Choice

$$\lim_{x\rightarrow\infty}\ \frac{2x-3}{4x\ +1}$$  

• 0

• DNE

• 1/2

• 2

Multiple Choice

Find the limit

• ⁶/₅

• 0

• $$\infty$$

• $$-\infty$$
  

Multiple Choice

Check the image

• 0
• 2
• 3
• 4

Multiple Choice

f(2) =

• 0

• 2

• 3

• 4

Multiple Choice

What is the limit of the function as x approaches -4 from the left?

• 2
• -4
• DNE
• -2

Multiple Choice

What is the limit of the function as x approaches 1 from the right?

• DNE
• 1
• 4
• -2