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 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 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
Lesson 3 - Continuity
Continuity at an Interior Point - A function y = f(x) is continuous at an interior point c of its domain if \lim_{x\to c}f\left(x\right)=f\left(c\right)
Continuity at an Endpoint - A function y=f(x) is continuous at a left endpoint a if \lim_{x\to a^{-}}f\left(x\right)=f\left(a\right) and at a right endpoint b if \lim_{x\to b^{+}}f\left(x\right)=f\left(b\right)
Types of discontinuities
Removable Discontinuity - The x-value either makes the function undefined, or it is independently defined, and the left and right sided limits are equal
Jump Discontinuity - As x approaches a value, the left and right-sided limits exists, but are different
Infinite Discontinuity - If the x-value makes the denominator zero, and is not found in the numerator, it makes the equation have a vertical asymptote at that value, and one or both one-sided limits are infinite
Oscillating Discontinuity - If the equation oscillates too much approaching an x-value to have a discernable limit, the limit does not exist
Lesson 4 - Rates of Change and Tangent Lines
Average Rate of Change - The amount of change divided by the length of the interval
m=\frac{f\left(x+h\right)-f\left(x\right)}{h} , where h is the difference between the first and second x-value
The average slope is the secant line between the two points
As h approaches zero, the difference between the x values becomes zero, and the slope becomes that of the curve of the graph (tangent line)
Slope of a Curve at a Point - The slope of y = f(x) at a point, or the derivative using 1st principles, can be represented by m=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}
To solve for the slope of the tangent/derivative using 1st principles, substitute the function into the limit for the slope
For f(x+h), substitute the whole function, and input (x+h) into all places where the argument x is
Expand, cancel, and reduce terms and factors
Substitute all available h’s with 0 and solve
If it is a constant, that is the slope
If it contains x variables, substitue the x value of a given point to find the slope at that point
To find the equation of the tangent line, input the slope of the tangent into the point-slope form of an equation of a line
Point-Slope Form of a Line - y-y_1=m\left(x-x_1\right), where y1 and x1 are the coordinates of a given x-value or point
Normal Line - The line perpendicular to the tangent line, where the slope is the negative reciprocal of the tangent line’s slope
To solve for the normal line equation, substitue the negative reciprocal of the tangent line’s slope for m in the point-slope form of a line