Limits Jeopardy!

Definitions

100

Name the discontinuity: 

The graph approaches infinity at the point of discontinuity

Infinite discontinuity 

caused by a vertical asymptote

200

What is a sign chart

A sign chart is a visualization used to find limits. 

A line is drawn with markings at each zero and can be used to find if an infinite limit is negative or positive

300

What is a Jump Discontinuity

A place where the left side limit and the right side limit of an equation do not match, so the graph cannot be continuous

400

What is the Squeeze Value Theorem

If g(x) ≤ f(x) ≤ h(x) for all x in some interval about C AND limg(x)=limh(x)=L

THEN limf(x)= L at x=c

500

3 Conditions for continuity at x=c

1. function is defined at x=c

2. right and left limits of c are equal (limf(c) exists

3. limf(x)=limf(x)=limf(x)=f(c)

     x-> c    x->c-   x->c+

Properties

100

How do you simplify? 

lim[f(x)g(x)]

as x approaches c

limf(x)·limg(x)

As x approaches c

200

Is the limit of a quotient equal to the quotient of the limits?

YES

300

TRUE OR FALSE? 

lim[b·f(x)]=b·limf(x)

as x approaches c

TRUE

400

TRUE OR FALSE? 

lim[f(x) ± g(x)] = limf(x)·limg(x) 

as x approaches c

FALSE

limf(x) ± limg(x) as x approaches c

500

SIMPLIFY TO SOLVE

lim[f(x)/g(x)]

as x approaches c

limf(x) / limg(x)

IF limg(x)≠0 as x approaches c

Infinite Limits

100

What happens to the 5 in the problem

lim(5+4x+x³)                                           x -> ∞

It is insignificant compared to the ∞ plugged in at x³ so it is dropped

200

What is a good way to visualize infinite limits?

Draw a sign chart!

300

What is BOSTON?

A method commonly used to discover the presence of horizontal asymptotes. It refers to the degree of polynomials in numerator and denominator                                         Bottom                                               y=O                                                              Same a/b                                                 Top                                                           O                                                             N

400

What is the limit of a function with a vertical asymptote at x=c as x approaches c from the left?

±∞

500

What is the limit of sin(x) as x-> ∞

Why?

Does Not Exist (DNE)

It is an oscillating function so it does not get close to any one number as x approaches ±∞

Examples

100

lim(x²+3)

x -> 1⁺

4

200

lim[(x²-6x+1)/(x+3)]

x -> -3⁺

∞

300

lim(5+4x+x³) 

x -> ∞

∞

400

find lim(sinx) as x-> π/2

1

500

My brother drives his car to school every day. Today, he went 60mph on Creedmor. Was his speed 45 at any point in time? Why?

Yes- His speed started at 0 and rose to 60. According to the intermediate value theorem, it must have passed through 45 mph at some point.

Examples

100

If limf(x)=3 as x->c, 

find lim[3f(x)]

9

200

If limf(x)=4 at x->2 and limf(x)=3 and x->3

find limf(x)² as x->2

16

300

find lim(2sinx⁄3x) as x->0

2/3

400

Is this function continuous?

            x²-2, x<2

f(x)=    x+3, x>2

No there is a jump discontinuity at x=2

500

Find the limit as x->2

lim|x-2|/x-4

|0|/-2

=0