Limits & Continuity

composite function: find f[g(x)] and g[f(x)]

for every X in the outer function, plug in the entire inner function.

limit of composite function by finding the composite

1. find the composite function

2. take the limit of the composite function

limit of composite function by finding the inner composite and plugging it into the outer function.

1. find the limit of the inner function

2. plug the value of the limit into the outer function

finding removeable (point) and nonremovable (infinite) discontinuities on rational functions.

1. if any x-values can be factored out of the rational function, a removeable point discontinuity exists (negative value of the factor)

2. any x-values that cannot be factored out and make the function divided by 0 are nonremovable discontinuities: on a graph this is a vertical asymptote

3. define the any removeable point discontinuities by a piecewise function

intermediate value theorem

• if a function is continuous on a closed interval of x-values [a, b] and takes y values f(a)=A and f(b)=B, then it also takes every y value between A and B at least once the interval [a, b].

• if A is negative and B is positive, then there exists at least point c in the interval [a, b] such that y value f(c) = 0.

conjugate method to solve limits

1. multiple the num and denom by the conjugate of the complex two-term portion (either n or d)

2. solve the remaining function by canceling factors in the numerator and denominator.

3. substitue solutions into the equation to find the limit.

rules for finding horizontal asymptotes

1. degree N < D: y=o

2. degree N > D: HA does not exist

3. degree N = D: HA given by ratio of coefficients on highest degree (N/D)

trigonometric limit values

trigonometric Pythagorean theories

finding a value to make a function continuous

1. set the two piecewise functions equal to eachother

2. substitue any x values with the break point

3. solve the equation for the remaining variable

4. ensure the LH and RH limits are equal by plugging the break point into the two piecewise functions with the constant variable now defined to create a general limit.