Chapter 2: Limits and Continuity

Question 1

Week 2

State Galileo’s law of falling bodies.

Answer 1

Galileo’s law of falling bodies states that all bodies fall at the same rate when dropped from the same height provided air resistance is neglected.

In other words, in the absence of air resistance, a brick and a feather will hit the ground simultaneously if dropped from the same height.

He observed that free falling bodies follow a proportional relationship, which can be represented as:

• y ∝ t²
• y = kt²
• k = 16 feet/sec²

Therefore, y = 16t²

Question 2

Week 2 / Week 3

Define average speed and the average rate of change.

Answer 2

Suppose a moving object has traveled a distance x(t) in time t

The object’s average speed during the interval [t₁ - t₂] can then be defined as:

Avg. speed = x(t₂) - x(t₁)/ t₂ - t₁

The Average Rate of Change

The average rate of change of y=f(x) with respect to x over the interval [x₁, x₂] is:

Δy/Δx = f(x₂) - f(x₁) / x₂ - x₁

Let’s Slightly Tweak the Formula

Let’s assume that we want to calculate the average rate of change over the interval [x, x+h], with h being a constant interval between x₁ and x₂.

Δy/Δx = f(x+h) - f(x) / (x+h) - x

Δy/Δx = f(x+h) - f(x) / h

This formula can be used to calculate the average rate of change of the function over a constant interval.

Definition of the Average Rate of Change

The average rate of change between two points is the slope of the secant line between the two points.

Question 3

Week 2

Define instantaneous speed.

Answer 3

Instantaneous speed is an object’s exact speed at any moment in time.

It’s formula can be given by the following:

v = dx/dt.

For example, let’s assume that the function x(t) = 5t + 6 describes the displacement of an object.

The derivative of x(t) is x’(t) = v(t) = 5 m/s.

v(t) tells us that the velocity of the object at any given time is 5 m/s.

The instantaneous velocity of an object can also be defined using limits.

Let’s define v(t) as Δx/Δt.

The instantaneous velocity of a function is the limit of v(t) as Δt approaches 0.

Question 4

Week 3

What is the point-slope equation?

Answer 4

y - y₁ = m(x - x₁)

OR

y = mx + b

The former is more convenient since there is no need to compute b in order to reach the final form of the equation.

Question 5

Week 3

What is a limit?

Answer 5

What is a limit?

In mathematics, a limit, denoted as lim_x–>10 f(x) describes how a function f(x) behaves as x approaches a certain point a (10 in this case). It signifies the value that f(x) approaches as xgets closer to a

Limits are essential for calculus and understanding function behavior and continuity.

The limit of a function does not depend on how the function is defined at the point being approached since it never reaches said point.

What Methods Can Be Used to Compute Limits?

There exist various techniques that make limits relatively easy to compute. Each method is unique to its use-case.

Listed below are some methods that are commonly used to compute limits:

• Multiplying by conjugates
• Multiplication of the numerator and denominator by the highest power in the denominator.
• Factoring and simpliifying.
• expanding and simpifying
• L’hôpital Rule
• Adding elements while maintaining equality (applicable for limits involving sin( x ) / x)

Question 6

Week 3

What is the limit of different functions as x approaches a constant c

Answer 6

Limits in Identity Functions

If ‘f’ is the identity function f(x)=x, then, for any value of c, lim_x—>c f(x) = lim_x—>c x

Constant Functions

If f is a constant function f(x) = k, then for any value c

lim_x—>c f(x) = lim_x—>c k

Unit Step Functions

If a function is not continuous at point c, then its limits cannot be defined.

lim_x—>c+ f(x) ≠ lim_x—>c- (fx), the limit does not exist

Question 7

Week 3

List the properties of limits

Answer 7

If L, M, c, & k are real numbers and:

lim_x—>c+ f(x) = L

lim_x—>c+ g(x) = L

The Addition Property of Limits

**lim_x—>c+ (f(x) + g(x)) = L + M

The Difference Property of Limits

**lim_x—>c+ (f(x) - g(x)) = L - M

The Product Rule of Limits

**lim_x—>c+ (f(x) * g(x)) = LM

The Quotient Rule of Limits

**lim_x—>c+ (f(x) / g(x)) = L / M

g(x) ≠ 0

The Constant Multiple Rule of Limits

lim_x—>c+ k(f(x) = kL

The Power Rule of Limits

lim_x—>c+ f(x)ⁿ = Lⁿ

n > 0

The Root Rule of Limits

lim_x—>c+ f(x)^1/n = L^1/n

n > 0

Question 8

Week 4

What is a continuous function?

Answer 8

Continuity

A continuous function is a mathematical function that exhibits smooth, unbroken behavior without abrupt jumps, holes, or discontinuities in its graph.

This means that as you move along the function’s domain, even small changes in the input result in small, gradual changes in the output, ensuring a continuous curve without interruptions.

If a function f(x) is continuous, there exists no input value of f(x) that results in an undefined output, provided that the input values are within the domain.

Question 9

Week 4

What are the properties of additive continuous functions?

Answer 9

If f and g are continuous at x = c, the, the following algebraic combinations are continuous:

• Sum ( f + g)
• Difference ( f - g)
• Constant Multiples ( k * f(x))
• Product ( f * g)
• Quotient (f / g)
• Powers (fⁿ) ; n is an integer
• Root (ⁿ√f(x)) ; if it is defined on an interval containing c

Question 9

Week 4

Is every polynomial function continuous?

Answer 9

Yes

Question 10

Week 4

When are composite functions continuous?

Answer 10

If f is continuous at c and g is continuous at f(c), g(f(x)) is continuous at c.

Question 11

Week 4

When is a function continuous?

Answer 11

A function f(x) is continuous at point c when its right hand limit is equal to it’s left hand limit, and f(c) i.e. the limit exists.

lim ^{(x -> c+)} f(x) = lim ^{(x-> c-)} f(x) = f(c)

Question 12

Week 4

Define Intermediate Value Theorem (IVT).

Answer 12

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus. It states that if a continuous function f(x) takes on different values at two points a and b within a closed interval [a, b], then there exists at least one point c in that interval where f(c) equals k.k lies anywhere between f(a) and f(b).

This theorem guarantees the existence of solutions or roots for continuous functions.

Question 13

Week 4

Explain Continuous Extension.

Answer 13

If function f has a removable discontinuity at c and lim_{(x -> c)} exists and is some arbitrary value L, then we can define another function F.

F will be equal to f everywhere except point c, where it will be equal to L.

Function F is called the continuous extension of f.

Question 14

Week 4

State the rule of limits for rational functions.

Answer 14

Consider f(x) a rational, continuous function. if lim (x -> ±∞) f(x), divide the numerator and denominator of the rational function by the highest power of x in the denominator.

The value obtained after finding the limit of the function is the horizontal asymptote of f(x).

Question 15

Week 4

What is the sandwich theorem?

Answer 15

The Sandwich Theorem

Suppose that g(x)<=f(x)<=h(x). For all x values in some open interval containing c except possibly at x = c itself. Suppose also that.

lim_{x -> c} g(x) = lim_{x -> c} h(x) = L

Then,

lim_{x -> c} f(x) = L

Proving Special Limits using the Sandwich Theorem

lim_{x -> 0} sin( x ) / x

we can squeeze sin( x ) / x in between two slices of bread (functions)

cos (x) ≤ sin ( x ) / x ≤ 1 / cos( x )

we then try to find the limits of both bread slices (functions)

lim_{x -> 0} cos( x ) = 1

lim_{x -> 0} 1 / cos( x ) = 1

**since both functions that f(x) is sandwiched between converge at y = 1 when x approaches 0, we can assume that: **

lim_{x -> 0} sin ( x ) / x = 1

Special Inequalities

• |θ| <= sinθ <= |θ|
• |θ| <= - cosθ <= |θ|

The Sandwich Theorem may also be used to compute limits of trigonometric functions as x approaches infinity

Question 16

Week 5

Explain one-sided limits.

Answer 16

A one-sided limit is the value the function approaches as the x-values approach the limit from one side only.

• lim_{x -> c+} f(x) (RH Limit)
• lim_{x -> c-} f(x) (LH Limit)

Relation between one-sided limits and continuity

A limit only exists if the RH Limit is equal to the LH Limit.

A function is continuous at any point c when the limit of f(x) as x approaches c exists, and is equal to f(c)

Question 17

Week 5

Explain vertical asymptotes in the context of limits.

Answer 17

Vertical Asymptotes

Let’s assume that function f(x) is a function defined for all values of x except c.

As x approaches a value c, f(x) reaches infinity (positive or negative)

A line x=c would a vertical asymptote of the graph of a function y = f(x) if either,

lim_{x -> a-} f(x) = ±∞

or,

lim_{x -> a+} f(x) = ±∞

Question 18

Explain horizontal asymptotes in the context of limits.

Answer 18

Horizontal Asymptotes

Let’s assume that function f(x) is a function defined for all values of f except c.

As x approaches positive/negative infinity , f(x) converges upon a single point. (positive or negative)

Hence, line y = c would a horizontal asymptote of the graph of a function y = f(x) if either,

lim_{x -> + ∞} f(x) = c

or,

lim_{x -> - ∞+} f(x) = c

Question 19

Explain oblique asymptotes in the context of limits.

Answer 19

Oblique Asymptotes

An oblique asymptote, also known as a slant asymptote, occurs when a rational function’s degree of the numerator is one greater than the degree of the denominator. To find the oblique asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient of this division represents the oblique asymptote.

Example:

Find the oblique asymptote of the function f(x) = (x^2 + 3x + 5) / (x + 2).

Perform polynomial long division or synthetic division:

(x + 2) | (x^2 + 3x + 5)

Note: while doing polynomial long division, do not include the remainder in your answer for the equation of the slant asymptote

The result is x + 1.

The oblique asymptote is given by y = x + 1.

So, the oblique asymptote for f(x) is y = x + 1.