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Instantaneous Rate

The rate of y to x for a given point along the x axis

Secant Line

a line that intersects a circle at two points on the same plane

Tangent Line

Another name for a linear approximation to the curve is a

limit

The unique value that a function approaches as x-values of the function approach c from the left and right sides

Limit Notation

(the arrow notations is a subscript under lim that I can't do here)

Two-sided limit

One in which the limit is the same whether it is approached from above or below

One-sided limit

a limit that differs depending whether it is approached from above or below

Limit Laws: Sum

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Difference

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Constant Multiple

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Product

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Quotient

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Power

Assume lim f(x) and lim g(x) exist for x→a

Limit Laws: Fractional Power (or, Root)

Assume lim f(x) and lim g(x) exist for x→a

Squeeze Law

If f(x)≤g(x)≤h(x) for values of x near a, except possibly at a, then

Infinite Limit

If f is defined for all x near a, and f(x) grows arbitrarily large for all x sufficiently close (but not equal) to a, then

Limits at infinity

The dependent variable approaches a finite number as the independent variable becomes arbitrarily large

f(x)=1/x²

Example: Infinite Limit v Limits at Infinity

Asymptote

An imaginary line on a graph that acts as a boundary line.

vertical asymptote

comes from a factor in the denominator that doesnt cancel out

horizontal asymptote

a line that the curve approaches as x goes to infinity but never reaches

Infinite Limits at Infinity

If f(x) becomes arbitrarily large as x becomes arbitrarily large, then lim x→∞ f(x)∞

End Behavior

Look at HA

End Behavior of Powers and Polynomials

Find the term with x with the highest power

End Behavior of Rational Functions

Divide numerator and denominator by x^n where n is the largest power applied to x in the denominator

Values oscillate between ∓1 as x→∓∞

End Behavior of sin & cos x

Continuity

Refers to whether a function contains holes, jumps, or breaks

Continuity at a point

A function f is continuous at a if lim x→a f(x) = f(a)

Continuity Checklist

In order for f to be continuous at a

Continuity & Composite Function Rules

If functions f and g are continuous at a, then the following are also continuous at a, assuming that c is a constant and n > 0 is an integer

Continuity on an Interval

A function is continuous on an interval if it is continuous at every point in that interval.

[f(x)]^n/m

Continuity of Functions with Roots

Intermediate Value Theorem

If f(a) > 0 and f(b) < 0 then there is a real zero between a and b

Removable

Discontinuity Classes

ͼ

Epsilon

σ

Lambda

Limit Proximity

lim x→a f(x)=L means that f(x) is near to L for all x near to a

f(−x) = f(x)

f(x) is an even function if

f(−x) = −f(x)

f(x) is an odd function if

Limit Existence Theorem: [limit as x→c of f(x)] = L if and only if

[limit as x→c⁺ of f(x)] = L and [limit as x→c⁻ of f(x)] = L

b

limit as x→c of b =

c*[limit as x→c of f(x)]

limit as x→c of c*f(x) =

limit as x→c of [f(x)*g(x)] =

[limit as x→c of f(x)] * [limit as x→c of g(x)]

limit as x→c of [f(x)/g(x)] =

[limit as x→c of f(x)] / [limit as x→c of g(x)]

[limit as x→c of f(x)]ⁿ

limit as x→c of [f(x)]ⁿ =

0

[limit as θ→0 of (sin θ)/θ] =

1

[limit as θ→0 of (1 − cos θ)/θ] =

[limit as x→c of f(x)] = f(c)

f(x) is continuous at x = c if