MCV4U - Chapter 2 - Limits and Continuity
Communication
Limit notation must be written for each line until x is evaluated, either through substituting c or replacing the limit with a known value
Therefore statements are not required for finding limits, but they are required for finding the horizontal asymptote(s) equations
Lesson 1 - Rates of Change and Limits
Limit - The y-value of a function that the values of a function approach or equal as the x-values approach a specified value
\lim_{x\to c}f\left(x\right)=L
f(x) is the function
c is some specified x-value
L is the y-value limit
The function does not have to be defined at c to exist
\lim_{x\rightarrow0}\frac{\sin x}{x}=1 and \lim_{x\rightarrow0}\frac{\tan x}{x}=1 are specific values that can be used for substitution
Properties of limits:
\lim_{x\to c}k=k , where k is a constant
\lim_{x\to c}x=c
Sum Rule - \lim_{x\to c}\left(f\left(x\right)+g\left(x\right)\right)=\lim_{x\rightarrow c}f\left(x\right)+\lim_{x\rightarrow c}g\left(x\right)
Difference Rule - \lim_{x\to c}\left(f\left(x\right)-g\left(x\right)\right)=\lim_{x\rightarrow c}f\left(x\right)-\lim_{x\rightarrow c}g\left(x\right)
Product Rule - \lim_{x\to c}\left(f\left(x\right)\cdot g\left(x\right)\right)=\lim_{x\rightarrow c}f\left(x\right)\cdot\lim_{x\rightarrow c}g\left(x\right)
Constant Multiple Rule - \lim_{x\to c}\left(f\left(x\right)\cdot k\right)=\lim_{x\rightarrow c}f\left(x\right)\cdot k
Quotient Rule - \lim_{x\to c}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{\lim_{x\rightarrow c}f\left(x\right)}{\lim_{x\rightarrow c}g\left(x\right)} , as long as g(x) does not equal zero
Power Rule - \lim_{x\to c}\left(f\left(x\right)\right)^{n}=\left(\lim_{x\to c}\left(f\left(x\right)\right.\right)^{n}
Two-Sided Limit - The function approaches the same limit from both sides as x→c
If the function does not approach the same limit on both sides, it does not exist (DNE); however, it may have two distinct, real one-sided limits
Right-hand Limit - The limit as f of x approaches c from the right, denoted by x→c+
Left-hand Limit - The limit as f of x approaches c from the left, denoted by x→c-
A function only has a limit as x approaches c if both one-sided limits are equal\lim_{x\to c}f\left(x\right)\lrArr\lim_{x\rightarrow c^{+}}f\left(x\right)=L,\lim_{x\rightarrow c^{-}}f\left(x\right)=L
When given a piecewise function, x<c indicates approaching from the left side, and c<x indicates approaching from the right side
To determine a graph from a limit, there needs to be a continuous graph that the function can follow
To evaluate limits:
Use properties of limits
Substitution, as long as it does not create a zero denominator
Factoring
Sum of Cubes - a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)
Difference of Cubes - a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)
Rationalize the numerator or denominator by multiplying by the conjugates
Simplify fractions
Convert to sine and cosine
Use pre-defined rules
Use long division for cubic, quartic, and quintic functions
Lesson 2.2a - Limits Involving Infinity
x approaching infinity, ∞, means it moves to the right, and x approaching negative infinity, -∞, means it moves to the left
In the function f\left(x\right)=\frac{1}{x} :
\lim_{x\rightarrow\infty}\frac{1}{x}=0
\lim_{x\rightarrow-\infty}\frac{1}{x}=0
Horizontal Asymptote - The line y=b is a horizontal asymptote if the limit as x approaches either infinity or negative infinity equals b
Horizontal asymptotes can exist on one or both sides, and they can be crossed
To find a horizontal asymptote, use techniques for solving limits as x approaches infinity and/or negative infinity
If the function is a polynomial over a polynomial, only check one side, as the limit will be the same on both sides
For polynomials over polynomials, divide the numerator and the denominator by the highest degree of x present in the function
If the limit ends up being undefined, there is no horizontal asymptote
\lim_{x\rightarrow\infty}\frac{\sin x}{x}=0 , if x is any value
In absolute value functions, |x|, a graph can have more than one horizontal asymptote
To solve, evaluate as two separate limits, with a positive and negative x, respectively
When evaluating functions by order of size, a function determined to be \frac{big}{bigger} will have a horizontal asymptote at y=0
If the limit is of a function is positive or negative infinity, there is no horizontal asymptote
Order of Size:
\frac{1}{x} (smallest)
\log\left(x\right)
x
x^2
2^{x}
x^{x} (largest)
Lesson 2.2b - Limits Involving Infinity, Vertical Asymptotes, and End Behaviour
Vertical Asymptote - x=c is a vertical asymptote if either \lim_{x\rightarrow c^{+}}f\left(x\right)=\pm\infty or \lim_{x\rightarrow c^{-}}f\left(x\right)=\pm\infty
To find a vertical asymptote:
Check first to see if there are any removable discontinuities by factoring the numerator and denominator
Substitute a value slightly to the left of the x value that would make the function undefined into x, and evaluate
At each step, determine if this would give a small negative, a large negative, a small positive, or a large positive
Ultimately, large positives will equal positive infinity, and large negatives will equal negative infinity
The limits will indicate if there is a vertical asymptote, as well as the end behaviours
To sketch the whole graph, solve for a horizontal asymptote, as well as the y-intercept, where necessary