Calc Review Unit 1: Limits and Continuity

Limit

What value a function is approaching as the independent variable approach a specific value

Average Rate of Change

f(b)-f(a)/ b-a

-secant line

-smallest subset is most accurate

An estimate of the instantaneous rate of change

f(2) - f(1.999)/ 2-1.999

-specific point

-tangent line

True or False

A limit shares what f(x) is approaching, not the value of f(x)

One-sided limit

The Y value a function approaches as you approach a given x-value from either the left or right side

What is the answer if two sides of ta limit are different

the limit as x approaches c does not exist

How many decimals do you round to during the AP test?

Three deciamls, round or truncate

What does lim {f(x)+f(x)}=

2lim f(x)

How do you approach limits within limits?

Start on the inside

Whats the first step to determining limits using algebraic manipulation

Direct substitution

Whats the second step to determining limits using algebraic manipulation (if Direct substitution does not work)

Factor and cancel out

-A limit does not exist when you factor as much as possible, and it would still equal zero

What if you factor as much as possible and the limit still equals zero?

The limit does not exist

What does lim of sin x / x as x approaches zero equal?

1

What does the limit 1-cosx/ x as x approaches 0 equal?

0

What does the limit x/ sin x as x approaches 0 equal?

1

What does the limit cos x -1/ x as x approaches 0 equal?

0

What does the limit sin3x/x as x approaches 0 equal?

3

How do you rationalize limits with fractions with rationals?

Multiplying by the conjugate, and using direct substitution

Whats the conjugate

(a-b) and (a+b) are conjugates

How do you rationalize limits with complex fractions?

Get a common denominator, multiply by the reciprocal, and use direct substitution.

Whats the squeeze theorem

If g(x) < f(x) , h(x), and if lim g(x) = L and lim h(x)=L, than lim f(x)= L, as x approaches a.

What functions typically can squeeze theorem not be used for

unbounded limits

How to find the value of a limit with absolute value?

Plug in a close value as x

|3.999-4|/3.999-4 =-1

What is a hole?

A removable discontinuity

What is a vertical asymptote?

a non removable discontinuity

What is a jump?

A non renewable discontinuity?

How to figure out discontinuities?

Factor, and Set the denominator equal to zero and solve

How can a hole discontinuity be found?

If the factors between the numerator and denominator cancel out

How can a VA be found?

If the factor doesn’t cancel out with the numerator

What discontinuities does x²-1 have?

none, parabolas are continuous and this function has no denominator

Whats the first condition for continuity

f of c is defined ( c exists in the domain)

Whats the second condition for continuity

lim f(x) exists (equivalent on both sides

Whats the third conditino for continuity

lim f(x) as x approaches c is equal to f of c

How to figure out a jump discontinuity?

if the limit from the left at c does not equal the limit from the right at c

Whats a domain restriction that has to do with the denominator

The denominator can not equal zero.

Whats a domain restriction that has to do with radicals

Radicals > 0.

Exapmple:

Square root of 7x-3. 7x-3 >0 x must be greater than 3/7

Whats a domain restriction that has to do with logarithms

The inside of a logarithm must be greater than zero

examplle:

log (2x+1), 2x+1>0, x must be greater than -1/2

What happens when you remove a dicontinuity?

You are filling in the hole

Notation of a hole discontinuity

lim f(x) exists, but it does not equal the value f(x)

How to find a hole

Identify the x-value, and find the y value using the limit

How do you find the y value using the limit

By plugging the x value into the function after removing the discontinuities

If the function f ix continuous for all real numbers and if f(x) = x²+6x+8/ x+4 when x can not equal -4, then f(-4) =

-2

What values do you use when evaluating infinite limits with vertical asymptotes?

Substitute close values for x

Definition of a Horizontal Asymptote

End behavior of a graph

-the graph can cross it unlike a VA

What does the limit of a horizontal asymptote equal if the denominator grows faster than the numerator?

0

What does the limit of a horizontal asymptote equal if the numerator and denominator grow equally

1

What does the limit of a horizontal asymptote equal if the numerator grows faster than the denominator

Infinity

Is there a horizontal asymptote if the numerator is growing faster?

no

Whats the first IVT justification

1) The function f(x) is continuous on the interval [a,b]

Whats the second IVT justification

f(a) can not equal f(b)

Whats the third justification for IVT

f (c ) or k is between f(a) and f(b)

IVT Conclusion

Therefore IVT applies and there exists a value c between a and b such that f (c ) = k.