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The limit as x approaches a of ƒ(x) exists if and only if

limit as x approaches a from the right of ƒ(x) is equal to the limit as x approaches a from the left of ƒ(x)

ƒ(x) is continuous at x=a if

limit as x approaches a of ƒ(x) is equal to ƒ(a)

Intermediate Value Theorem

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

Derivative

limit as h approaches 0 of {ƒ(x+h) - ƒ(x)}/h

A function is differentiable at x=a if

The function is continuous at x=a and the limit as h approaches 0 of {ƒ(x+h) - ƒ(x)}/h

limit as x approaches ±∞ of ƒ(x)=b

A function has a horizontal asymptote at y = b if

#/0

Vertical asymptotes occur at values of x that produce

0

Limit as x→∞ of (ax^m)/(bx^n), n>m

a/b

Limit as x→∞ of (ax^m)/(bx^n), n=m

increasing

ƒ'(x)>0

decreasing

ƒ'(x)<0

Relative (or Local) Maximum

ƒ'(x) changes from positive to negative

Relative (or Local) Minimum

ƒ'(x) changes from negative to positive

ƒ''(x)>0

Concave Up

ƒ''(x)<0

Concave Down

point of inflection

When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a

critical point

a point in the domain of the function at which the derivative is zero or undefined.

ln(ab)

ln(a)+ln(b)

ln(a/b)

ln(a) - ln(b)

aln(b)

ln(b^a)

0

Derivative: c

cnxⁿ⁻¹

Derivative: (cxⁿ)

1/(2√x)

Derivative: √2

Derivative: (ƒ(x)g(x))

ƒ(x)g'(x) + ƒ'(x)g(x)

(g(x)ƒ'(x)-g'(x)ƒ(x))/g(x)²

Derivative: (ƒ(x))/(g(x))

ƒ'(g(x))g'(x)

Derivative: (ƒ(g(x)))

ƒ'(x)e^ƒ(x)

Derivative: e^ƒ(x)

ƒ'(x)/ƒ(x)

Derivative: ln(ƒ(x))