Unit 1: Limits and Continuity

Question 1

What is the definition of a limit?

Answer 1

A limit is the value that a function approaches as the input approaches a certain point.

Question 2

True or False: The limit of a function can be different from the function’s value at that point.

Answer 2

True

Question 3

Fill in the blank: The notation for the limit of f(x) as x approaches a is ______.

Answer 3

lim(x→a) f(x)

Question 4

What is the Squeeze Theorem used for?

Answer 4

It is used to find limits of functions that are squeezed between two other functions.

Question 5

What does it mean for a limit to be infinite?

Answer 5

It means that the function increases or decreases without bound as it approaches a certain point.

Question 6

What is the formal definition of continuity at a point?

Answer 6

A function is continuous at a point if the limit as x approaches the point equals the function’s value at that point.

Question 7

Multiple choice: Which of the following is a requirement for a function to be continuous at x = c? A) f(c) exists B) lim(x→c) f(x) exists C) lim(x→c) f(x) = f(c) D) All of the above

Answer 7

D) All of the above

Question 8

What is an asymptote?

Answer 8

An asymptote is a line that a graph approaches but never touches.

Question 9

True or False: Vertical asymptotes occur where a function approaches infinity.

Answer 9

True

Question 10

What is the difference between a removable and a non-removable discontinuity?

Answer 10

A removable discontinuity can be ‘fixed’ by redefining the function at that point, while a non-removable discontinuity cannot be fixed.

Question 11

What is the Intermediate Value Theorem?

Answer 11

The IVT states that if f is continuous on the interval [a,b] and N is a value between f(a) and f(b), then there exists at least one c in (a,b) such that f(c)=N.

Question 12

Multiple choice: If a function has a limit of 0 as x approaches 2, what can we conclude? A) f(2) = 0 B) f(x) approaches 0 as x approaches 2 C) f(2) does not exist D) All of the above

Answer 12

B) f(x) approaches 0 as x approaches 2

Question 13

What does it mean for a limit to exist?

Answer 13

A limit exists if the left-hand limit and the right-hand limit as 𝑥 approaches a point 𝑐 are equal.

Question 14

Define a one-sided limit.

Answer 14

A one-sided limit is the value a function approaches as the input approaches a specific point from either the left or right.

Question 15

What are the three types of discontinuities?

Answer 15

1. Removable Discontinuity (hole in the graph)
2. Jump Discontinuity (sudden jump in values)
3. Infinite Discontinuity (vertical asymptote)

Question 16

Squeeze Theorem

Answer 16

f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them

Question 17

How do you evaluate limits at infinity?

Answer 17

To evaluate limits at infinity, analyze the end behavior of the function as x→∞ or x→−∞ by comparing the degrees of the terms in the numerator and denominator (for rational functions).

Question 18

What is the difference between horizontal and vertical asymptotes?

Answer 18

Horizontal asymptotes occur as x→±∞ and indicate the end behavior of a function.

Vertical asymptotes occur at values of 𝑥 where the function approaches infinity or negative infinity, typically where the denominator is zero in a rational function.

Question 19

State the Constant Multiple Law for limits.

Answer 19

If lim x→c f(x)=L and 𝑘 is a constant, then lim x→c k⋅f(x)=k⋅L

Question 20

What is the Sum Law for limits?

Answer 20

If lim x→c f(x) = L and lim x→c g(x) = M, then lim (f(x) + g(x)) = L + M

Question 21

What is the Difference Law for limits?

Answer 21

If lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c (f(x) − g(x)) = L − M

Question 22

What is the Constant Multiple Law for limits?

Answer 22

If lim x→c f(x) = L and k is a constant, then lim x→c (k ⋅ f(x)) = k ⋅ L

Question 23

What is the Product Law for limits?

Answer 23

If lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c (f(x) ⋅ g(x)) = L ⋅ M

Question 24

What is the Quotient Law for limits?

Answer 24

If lim x→c f(x)=L and lim x→c g(x)=M, and M ≠0, then lim x→c f(x) / g(x) = L / M

Question 25

What is the Power Law for limits?

Answer 25

If lim x→c f(x) = L and a is a positive integer, then lim x→c ​(f(x))ª =Lª

Question 26

What is the Root Law for limits?

Answer 26

If lim x→c f(x) = L and a is a positive integer, then lim x→a ª√f(x) = ª√L, provided L≥0 when a is even

Question 27

lim x-> 0 sin ax/ax = 1

Answer 27

One of two special limits

Question 28

lim x-> 0 (1 - cos(ax)) / ax

Answer 28

One of two special limits

Question 29

Types or limit discontinuities

Answer 29

1. Jump Discontinuity 2. Infinite Discontinuity 3. Removable Discontinuity 4. Essential Discontinuity

Question 30

How to determine a Jump Discontinuity

Answer 30

the lim x->a - f(x) ≠ the lim a->a + f(x)

Question 31

How to determine a Infinite Discontinuity

Answer 31

Check if lim x→a −f(x) or lim x→a + f(x) tends to ∞ or −∞, if yes then there is an infinite discontinuity at x=a.

Question 32

How to determine a Removable Discontinuity

Answer 32

1. Check if the limit lim x→a f(x) exists 2. Check if the function is either not defined at x=a, or if it is defined but f(a) ≠ lim x→a f(x)

Question 33

How to determine a Essential Discontinuity

Answer 33

If the function oscillates wildly as x→a such as trigonometric functions

Question 34

Rationalizing the numerator

Answer 34

Multiple the numerator containing the root (ex. √(x+7) + 1) by its conjugate (√(x+7) - 1) to the num. and denominator

Question 35

When is rationalizing the numerator necessary

Answer 35

help simplify expressions when calculating limits, especially when dealing with indeterminate forms like 0/0

Question 36

What does 0/0 mean

Answer 36

Indeterminate form (M0RE / W0RK) like 1. Simplify the Expression (Algebraic Manipulation) 2. Factorization 3. Rationalization 4. Special limits 5. L'Hopital's Rule

Question 37

limit of a function as x approaches a number involves...

Answer 37

determining the value that the function approaches as x gets arbitrarily close to a particular point

Question 38

Steps to solve for a limit of a function as x approaches a number

Answer 38

1. Direct Substitution (if possible) 2. If = 0/0 then use factorization, simplification, Rationalization 3. Check for One-Sided Limits (if needed)

Question 39

limit as x approaches infinity involves...

Answer 39

determining the behavior of the function as x becomes arbitrarily large

Question 40

Steps to solve for a limit of a function as x approaches infinity

Answer 40

Determine the highest power of f(x) = p(x) / q(x) - if p(x) < q(x), then y = 0 - If p(x) = q(x), the horizontal asymptote is y = leading coefficient of num / leading coefficient of den - if p(x) > q(x), then the function may approach infinity or negative infinity (no horizontal asymptote)

Question 41

How to calucalate the limit of piecewise functions at a particular point

Answer 41

look at the left-hand limit and right-hand limit. If these two one-sided limits match, then the limit exists. If they do not match, the two-sided limit does not exist at that point.

Question 42

Steps to find piecewise limits

Answer 42

1. Identify the point of interest where you want to find the limit 2. Check lim x→a −f(x) 3. Check lim x→a +f(x) 4. Compare the left-hand and right-hand limits

Question 43

A piecewise function is continuous at a point c if...

Answer 43

The limit as x→c exists ( the left-hand limit and right-hand limit are equal). 2. The function value at x=c exists (f(c) is defined). 3. The limit equals the function value (lim x→c f(x)=f(c))

Question 44

Steps to Make a Piecewise Function Continuous

Answer 44

1. Find lim x→a −f(x) 2. Find lim x→a +f(x) 3. Set the function value x=c to be the same as the common limit 4. If the function has a gap or discontinuity, adjust the value of f(c) so that: lim x→c f(x)=f(c)

Question 45

Steps to find Find lim x→a −f(x)

Answer 45

1. Identify the function and the point of interest 𝑎 2. determine how the function behaves as x approaches 𝑎 from values smaller than 𝑎 (consider values like 𝑎−0.1,𝑎−0.01,...) 3. Check for continuity or discontinuity

Question 46

Steps to find Find lim x→a +f(x)

Answer 46

1. Identify the function and the point of interest 𝑎 2. determine how the function behaves as x approaches 𝑎 from values bigger than 𝑎 (consider values like 𝑎+0.1,𝑎+0.01,...) 3. Check for continuity or discontinuity

Question 47

Formula and steps for the Limit of a Composition

Answer 47

Lim x→a f(g(x)) = f(lim x→a g(x)) 1. Find the limit of the inner function g(x) as x→a 2. Substitue the value lim x→a g(x) into the function f(x)

Question 48

limit of the composition of functions refers to...

Answer 48

first find the limit of g(x) as x→a, and then evaluate the outer function f(x) at that limit.

Question 49

limit of the composition of functions is valid if...

Answer 49

the inner function g(x) approaches a value that is within the domain of the outer function f(x). If lim x→a g(x) is outside the domain of f(x), the limit does not exist.

Question 50

Average Rate Or Change is...

Answer 50

Slope between two points on a curve

Question 51

AROC Formula on [a,b]

Answer 51

[f(b) - f(a)] / (b - a)

Question 52

Instantaneous Rate of Change (IROC) is...

Answer 52

the slope of the line tangent to the curve at a single point (x=a)

Question 53

IROC Formula at x = a

Answer 53

m = lim h→0 (f(a + h) - f(a)) / h

Question 54

What are horizontal asymptotes in limits?

Answer 54

a horizontal asymptote indicates the end behavior of a function—what the function "settles" toward as x moves farther away from the origin (to infinity or negative infinity).

Question 55

A function f(x) has a horizontal asymptote at y=L if...

Answer 55

as x approaches ∞ or −∞, the function values approach the constant L

Question 56

Steps to find horizontal asymptotes for Rational Functions: If f(x) = q(x) / p(x) and both are polynomial

Answer 56

- if p(x) < q(x), the horizontal asymptote is y = 0 - If p(x) = q(x), the horizontal asymptote is y = leading coefficient of num / leading coefficient of den - if p(x) > q(x), then there i no horizontal asymptote (the function may approach infinity or negative infinity) (Same as finding limits with infinities)