Calculus I exam 1

means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, it can be stated that "the limit of f of x, as x approaches c, is L". Note that this statement can be true even if f(c) ≠ L. Indeed, the function f(x) need not even be defined at c.

• 1. f(x) approaches a different number from the right side of c than it approaches from the left side.
• 2. f(x) increases or decreases without bound as x approaches c
• 3. f(x) oscillates between two fixed values as x approaches c

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then lim x→a f(g(x)) = f (lim x→a g(x)) = f (b).

• Substituting 0 for x yields 5/0, which is meaningless; hence, = DNE. (Remember, infinity is not a real number.)
• THIS IS AN ASYMPTOTE, NON-REMOVABLE DISCONTINUITY (numerator ≠ 0 , denominator = 0)

• 1. Learn to recognize which limits can be evaluated by direct substituation
• 2. If the limit of f(x) as x approaches c cannot be evaluated by direct substituaiton, try to find a afunction g that agrees with f for all of x other than x=c
• 3. apply the theorem for removable discontinuities such that; lim _x→c f(x) = lim _x→c g(x) = g(c)
• 4. use a graph or table to reinforce conclusion

If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists

lim _x→c f(x) = lim _x→c g(x)

• 1. f(c) is defined
• 2. lim x→c f(x) exists
• 3.

• continuity on an open interval: a function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line ( , ) is everywhere continuous.

if and only if and

A function is said to be continuous on [a,b] if and only if

• 1. it is continuous on (a,b),
• 2. it is continuous from the right at a and
• 3. it is continuous from the left at b.

• 1. scalar multiple: bf
• 2. sum or difference: f (+ - ) g
• 3. Product: fg
• 4. quotient: f/g, if g(c) ≠ 0

If the function g is continuous at a number a, and f is continuous at g(a), the composite function is also continuous at a.

If a function f is continuous on a closed interval [a,b], then for every value k between f(a) and f(b) there is a value c between a and b such that f(c) = k.

We say if we can make f(x) arbitrarily large for all x sufficiently close to x=a, from both sides, without actually letting .

We say if we can make f(x) arbitrarily large and negative for all x sufficiently close to x=a, from both sides, without actually letting .

if f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f

• let h and g be continuous on an open interval containing c. If g(c) ≠ 0, h(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of the function given by:
• has a vertical asymptote at x = c

then

then

if f is defined on an open interval containing a, and if the limit: exists, then the lines passing through (a, f(a)) with slpoe m is the tangent line to the graph of f at the point (a, f(a))

the derivative of f at x

if f is differentiable at x = c, then f is continuous at f = c

the derivative of a constant function is 0. That is, if c is a real number, then

if n is a rational number, then the function f(x) = xⁿ is differentialble. For f to be differentiable at x=0 then n must be a number such that x^n-1 is defined on the interval containing 0

• The derivative of the sum of two functions is the sum of the derivatives of the two functions:
• .
• Likewise, the derivative of the difference of two functions is the difference of the derivatives of the two functions.

find the derivative f'(c) = m = slope

plug m into the point slope formula y-y₁=m(x-x₁)