Unit 1 - Limits Flashcards
Limit of a function with constant value k
lim(k) as x—>c = k
Limit of the identity function at x = c
lim(x) as x—>c = c
Sum Rule of limits
lim (f(x) + g(x)) as x—>c = L + M
Difference Rule of limits
lim (f(x) - g(x)) as x—>c = L - M
Product Rule of limits
lim (f(x)g(x)) as x—>c = LM
Constant Multiple Rule of limits
lim (kf(x)) as x —>c = kL
Quotient Rule of limits
lim (f(x)/g(x)) as x—>c = L/M, M doesn’t equal 0
Power Rule of limits
lim (f(x))^r/s as x—>c = L^r/s
Limit of polynomial f(x)
f(c)
Limit of the quotient of polynomials
lim (f(x)/g(x)) as x—>c = f(c)/g(c) where g(c) doesn’t equal 0
Squeeze Theorem, where g(x) is less than or equal to f(x) which is less than or equal to h(x) for x+c is an interval about c
lim g(x) as x—>c = lim h(x) as x—>c = L, then lim f(x) as x—>c = L
Limits describe…
How a function behaves as inputs approach a particular value
How do limits as x—>c depend on how the function is defined at c?
They don’t
Sum, difference, product, constant multiple, quotient, and power rules apply as limits approach infinity
True
Horizontal asymptote
lim f(x) as x—>+-infinity = b
Vertical asymptote
lim f(x) as x—> a^+ = +-infinity, or lim f(x) as x—>a^- = +-infinity
Right end behaviour model
lim f(x)/g(x) as x—>infinity = 1
Left end behaviour model
lim f(x)/g(x) as x—>-infinity = 1
End behaviour model
Both a left and right end behaviour model
The graph of a quotient will always have a vertical asymptote when the denominator equals 0
False
End behaviour models can also model
horizontal asymptotes
Continuous at a point c
lim f(x) as x—>c = f(c) OR
lim f(x) as x—>a^+ = f(a) or lim f(x) as x —>b^- = f(b)
Composites of continuous functions are
continuous
Algebraic combinations of continuous functions are
continuous
3 step continuity test
(1) f(a) is defined
(2) lim f(x) as x—>a exists
(3) lim f(x) as x—>a = f(a)
Slope of a line or derivative
m = lim as h—>0 of [f(a+h) - f(a)]/h
Sensitivity (definition and equation)
how one variable responds to a small change in another variable
lim as x—>0 of delta(y)/delta(x)
Secant line
goes through two points on a curve
Average rate of a change can be thought of as
slope of a secant line
Calculating tangent to a curve
(1) slope of a secant line through P and Q
(2) limit of secant slope as Q approaches P
(3) 2 = slope at P, tangent to curve at P is the line through P with this slope
