Limits Flashcards
Tangent
The tangent to a curve is a line that touches the curve and has the same direction as the curve at the point of contact
Secant
A secant line to a curve is a line that intersects (or ‘cuts’) the curve more than once.
the limit of the slopes of the secant lines as we move a second point closer to a set point P
The slope of the tangent line at P
How do you say lim(x→a) f(x) = L ?
“the limit of f(x), as x approaches a, equals L”
What point is not considered when finding the limit of f(x) at x = a ?
in finding the limit of f(x) at x = a we never consider x = a
When finding the limit of f(x) at x = a does the function need to be defined at x = a ?
f(x) need not even be defined when x = a.
A Limit
Suppose f(x) is defined when x is near the number a. Then we write lim(x→a) f(x) = L
How can the limits of a function be estimated?
The limit of a function can be estimating by making the values of f(x) arbitrarily close to L by restricting x to be sufficiently close to a (on either side of a) but not equal to a.
The left-hand limit
lim(x →a−) f(x)=L
The right-hand limit
lim(x →a+) f(x) = L
How do you say lim(x →a+) f(x) = L ?
the right-hand limit of f(x), as x approaches a is equal to L
OR
the limit of f(x) as x approaches a from the right is equal to L
How do you say lim(x →a−) f(x)=L ?
the left-hand limit of f(x), as x approaches a is equal to L
OR
the limit of f(x) as x approaches a from the left is equal to L
What are the conditions for the left and right hand limits required for the limit to exist?
lim(x →a−) f(x) = L = lim(x →a+) f(x)
What does lim(x →a) f(x) = ∞ mean?
the values of f(x) can be made arbitrarily large by taking x sufficiently close to a, but not equal to a.
Infinite limits
lim(x →a) f(x) = ∞
AND
lim(x →a) f(x) = -∞
The vertical line x = a is called a vertical asymptote of the curve y = f(x) if at least one of these statements is true
lim(x →a) f(x) = ∞ lim(x →a) f(x) = -∞ lim(x →a-) f(x) = ∞ lim(x →a-) f(x) = -∞ lim(x →a+) f(x) = ∞ lim(x →a+) f(x) = -∞
Limit sum law
lim(x→a) [f(x)+g(x)] = lim(x→a) f(x) + lim(x→a) g(x)
Condition for the limit laws
Limits lim(x→a) f(x) and lim(x→a) g(x) exist c is a constant n is a positive integer
Limit difference law
lim(x→a) [f(x) - g(x)] = lim(x→a) f(x) - lim(x→a) g(x)
Limit constant multiple law
lim(x→a) [cf(x)] = c lim(x→a) f(x)
Limit Product Law
lim(x→a) [f(x)g(x)] = lim(x→a) f(x) · lim(x→a) g(x)
Limit quotient law
lim(x→a) [f(x)/g(x)] = [lim(x→a) f(x)] / [lim(x→a) g(x)]
Condition for the limit quotient law
lim(x→a) g(x) does not equal 0
Limit power law
lim(x→a) [f(x)]^n = [lim(x→a) f(x)]^n
Limit root law
lim(x→a) nth root[f(x)]= nth root[lim(x→a) f(x)]
Limits Direct substitution property
If f is a polynomial and a is the domain of f, then
lim(x →a) f(x) = f(a)
Squeeze theorem
If f(x) ≤ g(x) ≤ h(x) when x is near a (except maybe at a) and
lim(x →a) f(x) = lim(x →a) h(x) = L
then
lim(x →a) g(x) = L
What does the squeeze theorem mean?
if g(x) is squeezed between f(x) and h(x) near a ,and if f and h have the same limit L at a, then g is forced to also have the same limit L at a.
What makes a function continuous at a number a?
lim(x →a) f(x) = f(a)
What makes a function discontinuous at a?
It is not continuous at a
