Unit 1: Limits and Continuity Flashcards
What is the definition of a limit?
A limit is the value that a function approaches as the input approaches a certain point.
True or False: The limit of a function can be different from the function’s value at that point.
True
Fill in the blank: The notation for the limit of f(x) as x approaches a is ______.
lim(x→a) f(x)
What is the Squeeze Theorem used for?
It is used to find limits of functions that are squeezed between two other functions.
What does it mean for a limit to be infinite?
It means that the function increases or decreases without bound as it approaches a certain point.
What is the formal definition of continuity at a point?
A function is continuous at a point if the limit as x approaches the point equals the function’s value at that point.
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
D) All of the above
What is an asymptote?
An asymptote is a line that a graph approaches but never touches.
True or False: Vertical asymptotes occur where a function approaches infinity.
True
What is the difference between a removable and a non-removable discontinuity?
A removable discontinuity can be ‘fixed’ by redefining the function at that point, while a non-removable discontinuity cannot be fixed.
What is the Intermediate Value Theorem?
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.
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
B) f(x) approaches 0 as x approaches 2
What does it mean for a limit to exist?
A limit exists if the left-hand limit and the right-hand limit as 𝑥 approaches a point 𝑐 are equal.
Define a one-sided limit.
A one-sided limit is the value a function approaches as the input approaches a specific point from either the left or right.
What are the three types of discontinuities?
- Removable Discontinuity (hole in the graph)
- Jump Discontinuity (sudden jump in values)
- Infinite Discontinuity (vertical asymptote)
Squeeze Theorem
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
How do you evaluate limits at infinity?
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).
What is the difference between horizontal and vertical asymptotes?
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.
State the Constant Multiple Law for limits.
If lim x→c f(x)=L and 𝑘 is a constant, then lim x→c k⋅f(x)=k⋅L
What is the Sum Law for limits?
If lim x→c f(x) = L and lim x→c g(x) = M, then lim (f(x) + g(x)) = L + M
What is the Difference Law for limits?
If lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c (f(x) − g(x)) = L − M
What is the Constant Multiple Law for limits?
If lim x→c f(x) = L and k is a constant, then lim x→c (k ⋅ f(x)) = k ⋅ L
What is the Product Law for limits?
If lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c (f(x) ⋅ g(x)) = L ⋅ M
What is the Quotient Law for limits?
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
What is the Power Law for limits?
If lim x→c f(x) = L and a is a positive integer, then lim x→c
(f(x))ª =Lª
What is the Root Law for limits?
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
lim x-> 0 sin ax/ax = 1
One of two special limits
lim x-> 0 (1 - cos(ax)) / ax
One of two special limits
Types or limit discontinuities
- Jump Discontinuity
- Infinite Discontinuity
- Removable Discontinuity
- Essential Discontinuity
How to determine a Jump Discontinuity
the lim x->a - f(x) ≠ the lim a->a + f(x)
How to determine a Infinite Discontinuity
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.
How to determine a Removable Discontinuity
- Check if the limit lim x→a f(x) exists
- Check if the function is either not defined at x=a, or if it is defined but f(a) ≠ lim x→a f(x)
How to determine a Essential Discontinuity
If the function oscillates wildly as x→a such as trigonometric functions
Rationalizing the numerator
Multiple the numerator containing the root (ex. √(x+7) + 1) by its conjugate (√(x+7) - 1) to the num. and denominator
When is rationalizing the numerator necessary
help simplify expressions when calculating limits, especially when dealing with indeterminate forms like 0/0
What does 0/0 mean
Indeterminate form (M0RE / W0RK) like
1. Simplify the Expression (Algebraic Manipulation)
2. Factorization
3. Rationalization
4. Special limits
5. L’Hopital’s Rule
limit of a function as
x approaches a number involves…
determining the value that the function approaches as x gets arbitrarily close to a particular point
Steps to solve for a limit of a function as x approaches a number
- Direct Substitution (if possible)
- If = 0/0 then use factorization, simplification, Rationalization
- Check for One-Sided Limits (if needed)
limit as x approaches infinity involves…
determining the behavior of the function as x becomes arbitrarily large
Steps to solve for a limit of a function as x approaches infinity
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)
How to calucalate the limit of piecewise functions at a particular point
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.
Steps to find piecewise limits
- Identify the point of interest where you want to find the limit
- Check lim x→a −f(x)
- Check lim x→a +f(x)
- Compare the left-hand and right-hand limits
A piecewise function is continuous at a point
c if…
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))
Steps to Make a Piecewise Function Continuous
- Find lim x→a −f(x)
- Find lim x→a +f(x)
- Set the function value x=c to be the same as the common limit
- If the function has a gap or discontinuity, adjust the value of f(c) so that: lim x→c f(x)=f(c)
Steps to find Find lim x→a −f(x)
- Identify the function and the point of interest 𝑎
- determine how the function behaves as x approaches
𝑎 from values smaller than 𝑎 (consider values like 𝑎−0.1,𝑎−0.01,…) - Check for continuity or discontinuity
Steps to find Find lim x→a +f(x)
- Identify the function and the point of interest 𝑎
- determine how the function behaves as x approaches
𝑎 from values bigger than 𝑎 (consider values like 𝑎+0.1,𝑎+0.01,…) - Check for continuity or discontinuity
Formula and steps for the Limit of a Composition
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)
limit of the composition of functions refers to…
first find the limit of g(x) as x→a, and then evaluate the outer function f(x) at that limit.
limit of the composition of functions is valid if…
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.
Average Rate Or Change is…
Slope between two points on a curve
AROC Formula on [a,b]
[f(b) - f(a)] / (b - a)
Instantaneous Rate of Change (IROC) is…
the slope of the line tangent to the curve at a single point (x=a)
IROC Formula at x = a
m = lim h→0 (f(a + h) - f(a)) / h
What are horizontal asymptotes in limits?
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).
A function f(x) has a horizontal asymptote at
y=L if…
as x approaches ∞ or −∞, the function values approach the constant L
Steps to find horizontal asymptotes for Rational Functions: If f(x) = q(x) / p(x) and both are polynomial
- 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)
