Unit 2: Limits Flashcards
Average Velocity
The average velocity over a time interval
Instantaneous Velocity
The velocity at a single instant of time
Secant Line
A line joining two points on a curve
Limit
As t1 approaches t0, the average velocities typically approach a unique number, which is the instantaneous velocity
Tangent Line
The unique line that the secant lines approach
Limit of a Function (Preliminary)
Suppose the function f is defined for all x near a except possibly at a. If f(x) is arbitrarily close to L (as close to L as we like) for all x sufficiently close (but not equal) to a, we write
lim f(x) = L
x - a
and say the limit of f(x) as x approaches a equals L
Right-Hand Limit
Suppose f is defined for all x near a with x > a. If f(x) is arbitrarily close to L for all x sufficiently close to a with x > a, we write
lim f(x) = L
x - a+
and say the limit of f(x) as x approaches a from the right equals L.
Left-Hand Limit
Supposed f is defined for all x near a with x < a. If f(x) is arbitrarily close to L for all x sufficiently close to a with x < a, we write
lim f(x) = L
x - a-
and say the limit of f(x) as x approaches a from the left equals L
Relationship Between One-Sided and Two-Sided Limits Theorem
Assume f is defined for all x near a except possibly at a. Then lim f(x) = L
x - a
if and only if lim f(x) = L and lim f(x) = L
x - a+ x - a-
Limits of Linear Functions
Let a, b, and m be real numbers. For linear functions f(x) = mx + b,
lim f(x) = f(a) = ma + b
x - a
Limit Laws
Assume lim f(x) and lim g(x) exist.
x - a x - a
The following properties hold, where c is a real number and m > 0 and n > 0 are integers.
- Sum: lim (f(x) + g(x)) = lim f(x) + lim g(x)
x - a x - a x - a - Difference: lim (f(x) - g(x)) = lim f(x) -
x - a x - a
lim g(x)
x - a - Constant Multiple: lim (cf(x)) = c lim
x - a x - a
f(x) - Product: lim (f(x)g(x)) = (lim f(x)) (lim
x - a x - a x - a
g(x)) - Quotient: lim (f(x))/lim (g(x)) = lim f(x)/
x - a x - a
lim g(x), provided lim g(x) does not equal
x - a x - a
0 - Power: lim (f(x))^n = (lim f(x))^n
x - a x - a - Fractional Power: lim (f(x))^n/m = (lim
x - a x - a
f(x))^n/m, provided f(x) > or equal to 0 for x near a if m is even and n/m is reduced to lowest terms
Limits of Polynomial and Rational Functions
Assume p and q are polynomials and a is constant
a) Polynomial Functions: lim p(x) = p(a)
x - a
b) Rational Functions: lim p(x)/q(x) = p(a)/q(a) provided q(a) does not equal 0
x - a
Limit Laws for One-Sided Limits
Assume m > 0 and n > 0 are integers
a) lim (f(x))^n/m = (lim f(x)^n/m)
x - a+ x - a+
provided f(x) > or equal to 0 for x near a with x > a, if m is even and n/m is reduced to lowest terms
b) lim (f(x))^n/m = (lim f(x))^n/m
x - a- x - a-
provided f(x) > or equal to 0 for x near a with x < a, if m is even and n/m is reduced to lowest terms
The Squeeze Theorem
Assume the functions f, g, and h satisfy f(x) < or equal to g(x) < or equal to h(x) for all values of x near a, except possibly at a. If lim f(x) = lim h(x) = L, then lim g(x)
x - a x - a x - a
= L
Infinite Limits
Suppose f is defined for all x near a. If f(x) grows arbitrarily large for all x sufficiently close (but not equal) to a, we write
lim f(x) = infinity
x - a
We say the limit of f(x) as x approaches a is infinity.
If f(x) is negative and grows arbitrarily large in magnitude for all x sufficiently close (but not equal) to a, we write
lim f(x) = -infinity
x - a
In this case, we say the limit of f(x) as x approaches a is negative infinity. In both cases, the limit does not exist
One-Sided Infinite Limits
Suppose f is defined for all x near a with x > a. If f(x) becomes arbitrary large for all x sufficiently close to a with x > a, we write lim f(x) = -infinity, lim f(x), =infinity
x - a+ x - a-
and lim f(x) = -infinity are defined
x - a-
analogously as the one-sided infinite limits
Vertical Asymptote
If lim f(x) = + or - infinity,
x - a
lim f(x) = + or - infinity
x - a+
or lim f(x) = + or - infinity,
x - a-
the line x = a is called a vertical asymptote of f.
Limits at Infinity
If f(x) becomes arbitrarily close to a finite number L, for all x sufficiently large and positive x, then we write
lim f(x) = L
x - infinity
Horizontal Asymptote
The line y = L of f good morning it’s called
Infinite Limits at Infinity
If f(x) becomes arbitrarily large as x becomes arbitrarily large, then we write
lim f(x) = infinity
x - infinity
Limits at Infinity of Powers and Polynomials
Let n be a positive integer and let p be the polynomial, where an cannot equal 0
p(x) = anx^n + an-1x^n-1 + … + a1x + a0
- lim x^n = infinity when n is even
x - + or - infinity - lim x^n = infinity
x - infinity - lim x^n = -infinity when odd
x - -infinity - lim 1/x^n
x - + or - infinity
= lim x^-n = 0
x - + or - infinity
- lim p(x) = + or - infinity
x - + or - infinity
depending on the degree of the polynomial and the sign of the leading coefficient an
Slant/Oblique Asymptote
The line that is described when the graphs of two functions approach each other as x- -infinity
End Behaviour and Asymptotes of Rational Functions
Suppose f(x) = p(x)/q(x) is a rational function where
p(x) = amx^m + am-1x^m-1 + … + a2x^2 + a1x + a0
q(x) = bnx^n + bn-1x^n-1 + … + b2x^2 + b1x + b0
with am cannot equal 0 and bn cannot equal 0
- If m < n, then lim f(x) = 0
x - + or - infinity
and y = 0 is a horizontal asymptote of f - If m = n, then lim f(x) = am/bn
x - + or - infinity
and y = am/bn is a horizontal asymptote of f - If m > n, then lim f(x) = infinity
x - + or - infinity
or -infinity, and f has no horizontal asymptotes - Assuming that f(x) is in reduced form (p and q share no common factors), vertical asymptotes occur at the zeros of q
