Notes Packet 1 - PreCal Review, Limits, Continuity.pdf

Name: _______________________________________ Period: _________ CALCULUS NOTES

UNIT 1 - REVIEW OF PRECALCULUS, LIMITS, AND CONTINUITY

Ex. Find the x- and y-intercepts and domain and range, and sketch the graph. No calculator. (a)

2 ₆

3

x x

y x

− − =

−

x-int.:

y-int.:

Domain =

Range =

__________________________________________________________________________________________ (b) y= x+3

x-int.:

y-int.:

Domain =

Range =

__________________________________________________________________________________________ (c) y= 25−x2

x-int.:

y-int.:

Domain =

Range =

__________________________________________________________________________________________ (d) y 3x

x

=

x-int.: y-int.:

Domain =

Range =

__________________________________________________________________________________________ Ex. Sketch the graph. No calculator.

( )

2

1, 0 1

2 , 0 3

x x

f x

x x

 − ≤

 = 

Ex. Find the asymptotes (horizontal, vertical, and slant) and intercepts, and sketch the graph. (a)

( )

2

2 3

4 x f x

x

= −

Horiz.: Vert.:

Slant:

x-int.:

y-int.:

__________________________________________________________________________________________ (b)

( )

2 ₂

1

x x

g x x

+ =

− Horiz.:

Vert.:

Slant:

x-int.:

y-int.:

________________________________________________________________________________________

Ex. Solve. Draw a number line graph (sign chart) for each problem to show how you got your answer. Noncalc.

(d) 2

2 9 20

0 9

x x

x

− +

≥

− (e)

3 ₃ 2 _{4 12 0}

x + x − x− >

________________________________________________________________________________________ Ex. Solve. Give exact answers in radians, 0≤ <x 2π. No calculator.

(a) sinx = 1/2 (b) cosx =

(c) tan2x+ 3 tanx=0 (d) cscx− =2 0

(e) 2 cos2x−9 cosx− =5 0 (f) 2 cos2 x+3sinx=3

(g) sin 2

( )

x +cosx=0 (h) cos 2

( )

x =sinx

(i) 2 cos 3 2 x   =  

  (j) 2 sin 3

( )

x − 3=0

Ex. Solve. Show all steps. Use your calculator where needed.

(a) e2x−3=75 (b) e2x−3ex−10=0

(c) ex+12e−x− =7 0 (d) 18 5 3

x

e + =

(e) ln 3

(

x−2

)

=5 (f) log 4

(

x+3

)

=2

(g) log₅

(

x+ +3

)

log₅

(

x−2

)

=log 14₅ (h) log₄

(

2x+ −1

)

log₄

(

x−2

)

=1

________________________________________________________________________________ Ex. The population P of bears in a national park after t years is modeled by the function

0.3

160

1 7 t

P

e−

=

+ .

(a) How many bears were there after two years?

Introduction to Limits

lim

𝑥𝑥→𝑎𝑎𝑓𝑓(𝑥𝑥) =𝐿𝐿

• “The limit of f(x), as x approaches a, equals L.”

• “As the x values are getting closer and closer to “ a ”, the graph/f(x)/y-value gets closer and closer to “ L ”

3 ways to evaluate a limit:

A) Numerically (with a table) B) Graphically C) Analytically/Algebraically

Hint: Whenever working with limits algebraically, 1st try to direct evaluate (i.e. plug in the value of x). If you get an undefined/indeterminate answer, then try to factor (or use some other algebra

technique/trick) and see if it simplifies.

Important Note: Continuity does not affect the limit. As long as the graph approaches that value from both sides, the limit exists. All three of these functions have a limit, as x approaches “a”, equal to “L.”

We will talk more about continuity later…

One-Sided Limits:

When trying to identify limits, the left-sided limit and the right-sided limit must match. If they do, the limit exists. If the two sides do not merge towards the same value, then the overall limit does not exist (DNE) at that x value. Note: It is also acceptable to say “no limit exists” or NLE.

Left-sided limit: Right-sided limit:

Finding Limits Graphically and Numerically

Ex. Use your calculator to complete the table, and then use the results to estimate the limit. Then graph the function to confirm your results.

3

4 1 3

lim x

x x

→−

+ − =

+

x −3.1

−

3.01 −3.001 −2.999 −2.99 −2.9

( )

f x

__________________________________________________________________________________ Ex. Use your calculator to create a table, and then use the results to estimate the limit.

Then graph the function to confirm your results.

( )

0

sin 4

lim

x

x x

→ =

x

( )

f x

__________________________________________________________________________________ Ex. Sketch the graph, and then use it to find the limit.

(a)

( )

2

1, 2 3, 2

x x

f x

x

 + ≠

 = 

= 

( )

2

lim

x→ f x =

(b)

3

3 3

lim

x

x x

→−

−4 −3 −2 −1 1 2 3 4 5 6 7 8 9

−4 −3 −2 −1 1 2 3

x Ex. Use the graph of f shown on the right

to decide whether the value of the given quantity exists. If it does, find it. If not, explain why.

(a) f

( )

0 =

(b) f

( )

1 =

(c)

( )

0

lim

x→ f x =

(d)

( )

1

lim

x→ f x =

(e)

( )

3

lim

x→− f x =

(f)

( )

5

lim

x→ f x =

Evaluating Limits Analytically

Properties of Limits:

If L, M, c, and k are real numbers and lim

( )

and lim

( )

x→c f x =L x→cg x =M, then

1. Sum Rule: lim

(

( )

( )

)

x→c f x +g x = +L M

2. Difference Rule: lim

(

( ) ( )

)

x→c f x −g x = −L M

3. Product Rule: lim

(

( ) ( )

)

x→c f x ⋅g x = ⋅L M

4. Quotient Rule:

( )

( )

lim where 0

x c

f x L

M

g x M

→   = ≠      

5. Constant Multiple Rule: lim

(

( )

)

x→c k f x⋅ = ⋅k L

6. Power Rule: If r and s are integers, s≠0, then lim

(

( )

)

r r s s

x→c f x =L provided that

_Lrs _{is a real number.}

7. Limit of a Composite Function Rule: If f and g are functions such that

lim

( )

and lim

( )

( )

,

x→cg x =L x→cf x = f L then xlim→c f g x

(

( )

)

f limx→cg x

( )

f L

( )

.

 

= _{ }=

 

Ex. Given f x

( )

= −x 5, g x

( )

=x2. Evaluate the following. (a)

( )

1

lim

x→ f x =

(b)

( )

5

lim

x→ g x =

(c)

(

( )

)

1

lim

x→ g f x =

______________________________________________________________________________ Ex. Given that lim

( )

2 and lim

( )

3

x→a f x = x→ag x = , find the limit if it exists.

(a) lim 5

(

( )

)

x→a g x =

(b)

( )

( )

6 lim x a f x g x → + = ______________________________________________________________________________ Evaluate.

Ex.

(

2

)

2 3 5 1

lim

x→ x − x+ =

Ex.

5cos ₆

lim x x π →  _{ =}    

Ex.

( )

NOTE: If we cannot evaluate a limit by direct substitution, sometimes we can use algebraic techniques to rewrite the function so that we can evaluate the limit. No calculator.

Ex. 2 3 6 3 lim x x x x →− + − = + _____________________________________________________________________________ Ex. 4 3 1 4 lim x x x → − − = − _____________________________________________________________________________ Ex. _____________________________________________________________________________ Ex. 0 1 1 2 2 lim x x x → − + ₌ _____________________________________________________________________________

How to Factor - Sum/Difference of Two Cubes

3 3

3 3

Hint : a b

Ex.

(

)

0

3 3

lim

x

x x x

x

∆ →

+ ∆ − = ∆

_________________________________________________________________________________

Ex.

(

)

2

0

5 25

lim

x x

x

→

− −

=

_____________________________________________________________________________

Ex. lim

𝑥𝑥→3+

1 𝑥𝑥−3

_____________________________________________________________________________

Ex.

(

)

2

3

1

3

lim

x→ _x− =

_________________________________________________________________________________

Ex. where k is a constant

_____________________________________________________________________________

Ex. lim

𝑥𝑥→5

|𝑥𝑥−5|

𝑥𝑥−5

_____________________________________________________________________________

Ex. lim

𝑥𝑥→5+

𝑥𝑥2_{|𝑥𝑥−5|}

Evaluating Limits Analytically

Use your calculator to graph y sinx x

= .

What do you think 0 sin lim x x x

→ equals?

__________________________________________________________________________________

Now graph y 1 cosx. x

− =

What do you think 0 1 cos lim x x x → − equals? __________________________________________________________________________________ Two special trig limits you must know:

0 sin lim θ θ θ → = 0 1 cos lim θ θ θ → − ₌

Ex.

( )

0 sin 5 lim 5 x x x → = __________________________________________________________________________________

Ex.

( )

0 sin 5 lim 4 x x x → = __________________________________________________________________________________

Ex.

( )

LIMITS INVOLVING INFINITY (AND ZERO)

Lets look at a graph of an example rational function f(x) = 2(x+1) x−4 Remember the asymptotes? You need to!

The Vertical Asymptotes don’t really cause us too much trouble, they are easy to identify just create a specific x-value where the function either shoots up to positive infinity or drops down to negative infinity. If the function behaves differently on either side, obviously NLE. But if the function behaves the same, then we have something called an infinite limit.

There is debate between whether or not infinite limits exists in the same way that numerical ones do. Generally, it’s safe to say that NLE if your function rises or falls in an unbounded manner (to ± ∞).

For our example, describe the 2 limits around the VA and determine whether or not an infinite limit exists.

The Horizontal Asymptotes are a little bit more difficult to deal with. First of all, it’s harder to remember the rules for determining whether or not the function has a HA. Then you’re always going to have to check BOTH SIDES of the graph. The END BEHAVIOR of the function is determined by the HA(s). The function can approach a specific y-value as x gets very large, or it can rise forever y ∞, or fall forever y −∞

The RIGHT END is where x ∞ and the LEFT END is where x −∞.

Each end can do its own thing, or they can both approach the same y-value. It all depends on the function. __________________________________________________________________________________

Weird conceptual stuff time: What is happening to these numbers, getting larger or smaller?

EX) 1, 1/2, 1/3, 1/5, 1/10, 1/100, 1/10000….

As the denominator gets larger, the fraction overall gets ____________

So when the denominator gets really, really, REALLY large (approaches infinity), the fraction overall is going to get closer and closer to have a value of ______.

EX) 1/100, 1/50, 1/20, 1/10, 1/5, 1/2, …..

As the denominator gets smaller, the fraction overall gets ____________

So when the denominator gets really, really, REALLY small (approaches zero), the fraction overall is going to get closer and closer to have a value of ______.

__________________________________________________________________________________

Ok, so now lets recap: note: the # represents ANY constant number

Strategies for solving indeterminate (a) limits to infinity and (2) limits to zero

(a) Limits to infinity (or negative infinity)

1 (long way) – what you want to do is find the highest poweredterm (the largest exponent) in the fraction overall and divide the entire fraction by that one specific term. The goal when doing this is to only have constants and terms with “x”s left over in the denominators.

2 (shorter way) – find the largest single term on the top and the largest single term on the bottom and work only with those two terms in a fraction – you can ignore everything else. Reduce and evaluate. 3 (shortest way) – Use you HA rules. Note: ONLY WORKS FOR RATIONAL FUNCTIONS!

Examples: Evaluate the following limits. No calculator.

1. lim

𝑥𝑥→∞ 𝑥𝑥2−1 5𝑥𝑥2₊₁

2. lim

𝑥𝑥→∞ 3𝑥𝑥+5 2𝑥𝑥2_−7𝑥𝑥

3. lim

𝑥𝑥→−∞

𝑥𝑥2−101 4𝑥𝑥+3

4. lim

𝑥𝑥→∞ 3−𝑥𝑥 4𝑥𝑥+1

5. lim

𝑥𝑥→−∞ 5−𝑥𝑥2

4𝑥𝑥+1

6. lim

x →∞(x2−x)

7. A) lim

𝑥𝑥→∞

√2𝑥𝑥2₊₁

(b) Limits to zero

1 (long way) – what you want to do is find the lowest poweredterm (the smallest exponent) in the fraction overall and divide the entire fraction by that one specific term. The goal when doing this is to only have constants and terms with “x”s left over in the numerators.

2 (short way) – find the smallest single term on the top and the smallest single term on the bottom and work only with those two terms in a fraction – you can ignore everything else. Reduce and evaluate.

Note: NO HA RULES FOR LIMITS TO ZERO. THOSE ARE ONLY FOR INFINITY (end behavior)

Examples: Evaluate the following limits. No calculator.

1.

2.

3.

4.

__________________________________________________________________________________

The Formal / Epsilon-Delta Definition of a Limit

Squeeze Theorem or Sandwich Theorem:

If h x

( )

≤ f x

( )

≤g x

( )

for all x in an open interval containing c, except possibly at c itself, and if

( )

( )

( )

lim and lim , then lim .

x→ch x =L x→cg x =L x→c f x =L

Ex. If 2≤ f x

( )

≤x2+2 for all x, find

( )

0

lim .

x→ f x Sketch a graph to illustrate.

__________________________________________________________________________________ Ex. Graph the given function and the equations y= x and y= − x in the same viewing

window on your calculator. Then use the Squeeze Theorem to find

( )

0

lim .

x→ f x

( )

1

sin f x x

x  

= _{ }

Continuity

Definition of Continuity

A function f is said to be continuous at x=c if and only if:

1)

2)

3)

Ex. Sketch a function f so that:

(a) f c

( )

is not defined (b) lim

( )

x→c f x does not exist

(c) f c

( )

is defined and lim

( )

x→c f x exists but (d) f is continuous at x = c lim

( )

( )

x→c f x ≠ f c

______________________________________________________________________________ If a function f is not continuous at x = c, the discontinuity may be one of three types:

1) point discontinuity 2) jump discontinuity 3) asymptotic discontinuity

A point discontinuity is said to be a removable discontinuity because the function can be redefined at the point in such as way as to make the function continuous there. Jump discontinuities and

Ex. 1) Find the value(s) of x at which the given function is discontinuous. 2) Identify each value as a point, jump, or asymptotic discontinuity 3) Identify each value as a removable or nonremovable discontinuity. 4) Sketch the graph.

(a)

( )

2 ₄

2 x f x

x

− =

− (b)

( )

1 3 f x

x

= −

(c)

( )

3 3 x f x

x

− =

− (d)

( )

2 if 1 2 if 1

x x

f x

x x

+ <



=  _{− >} 

(e) f x

( )

=x2+1 (f)

( )

₂

3 x f x

x x

Ex. Use the definition of continuity to prove that the function is discontinuous at x = 2 if a= −4. Sketch the graph of f to check your answer.

( )

7 if 2

1 if 2

ax x

f x

x

+ ≠



=  ₌



__________________________________________________________________________________ Ex. Use the definition of continuity to find k so that f will be continuous at x=2 given

( )

3 if 2 6 if 2

x x

f x

kx x

+ ≤



=  _{+ >}



Sketch the graph of f to check your answer.

__________________________________________________________________________________

Ex. Use the definition of continuity to find a and b so that f will be continuous everywhere given

( )

17 if 3 if 3 1 3 if 1

x

f x ax b x

x

− ≤ −

 

=_ + − < ≤

 _>

Intermediate Value Theorem

Since a person’s height is a continuous function of time, if a child had a height of 58 inches at age 11 and a height of 64 inches at age 14, then the child’s height took on every value between 58 inches and 64 inches for some time between the age of 11 and the age of 14. This is a simple example of an important theorem in Calculus called the Intermediate Value Theorem.

Intermediate Value Theorem

If: i) f x

( )

is continuous on the closed interval [a, b] ii) if f a

( )

≠ f b

( )

iii) if k is between f a

( )

and f b

( )

,

then there exists a number c between a and b for which f c

( )

=k.

In other words, a function y= f x

( )

that is continuous on a closed interval

[

a b,

]

takes on every value between f a

( )

and f b

( )

. If k is between f a

( )

and f b

( )

, then there is at least one value c in

(

a b,

)

for which f c

( )

=k.

_________________________________________________________________________________________ Ex. Use the Intermediate Value Theorem to show that f x

( )

=x3−3x takes on the value 1 for

some x in the interval

[

−2, 4

]

.

_________________________________________________________________________________________ Ex. (a) Determine if the Intermediate Value Theorem holds for the given value of k.

(b) If the theorem holds, find a number c for which f c

( )

=k. If the theorem does not hold, give the reason.

(c) Draw a sketch of the curve and the line y=k.

( )

1 1 1

[ ] 2 7

2 2 4

f x a b k

x

= = =

−

 

 

 