Chapter 1 Test Limits

Limits that fail to exist

-Unbounded Behavior- Asymptotes of any kind, could be going in different directions -Behavior that differ from the right and left -Oscillating behavior- like an ekg, has no significant behavior and therefore doesn't approach anything so no limit.

What are the three ways to evaluate a limit?

1) Graphically 2) Algebraically 3) Numerically

What are the three types of discontinuity?

1) Removable 2) Jump 3) Infinite(asymptotic)

Strategy for Finding Limits

1. Learn to recognize which limits can be evaluated by direct substitution. 2. If the limit of f(x) as approaches c cannot be evaluated by direct substitution, try to find a function g that agree with f for all x other than x = c. 3. Prove it analytically. 4. Use a graph or table to reinforce your conclusion.

A function f is continous at c when these three conditions are met.

1. The function f(c) is defined. 2. The limit of f(x) exists at f(c). 3. The limit of f(x) exist at f(c), and is equal to f(c).

Limit of Trigonometric Functions

1. lim sin x = sin c 2. lim cos x = cos c x->c x->c 3. lim tan x = tan c 4. lim cot x = cot c x->c x->c 5. lim sec x = sec c 6. lim csc x = csc c x->c x->c

How to determine discontinuity of a piecewise function

Determine the limit of where there is a possible discontinuity (where the the separate functions meets)

How to simplify a function to an equation so they at all but one point

Factor Rationalize the numerator Simplify complex fraction

One-sided limits

For values evaluated from the left (negative superscript) and right (positive superscript) side respectively

Limit of a Composite Function

If f and g are functions such that lim g(x) = L and lim f(x) = f(L), x->c x->L then lim f(g(x)) = f(lim (g(x)) = f(L) x->c x->c

Limit

If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c is L

The Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) for all x in an open interval containing,(except possibly c itself) and if lim x ->c h(x)=L=lim x->c g(x) then lim x->a f(x) = lim x->a h(x) = L then lim x->a g(x)= L

Squeeze Theorem

If h(x)<_ f(x)<_ g(x) for all ox in an open interval containing c, except possibly at c itself and if lim h(x) = L = lim g(x) x->c x->c then lim f(x) exists and is equal to L. x->c

Limits of Polynomial and Rational Functions

If p is a polynomial function and c is a real number then lim p(x) = p(c) x->c If r is a rational function given by r(x) = p(x)/q(x) and c is a real number such that q(c) not equal to 0, then lim r(x) = r(c) = p(c)/q(c) x->c

Are rational and radical functions continuous?

In their domain they are continuous, but on an open interval no.

What happens if the one-sided portions of a limit do not match? That is the curve approaches different values depending on direction.

It does not exist as it causes a discontinuity.

Limit of a Function Involving a Radical

Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even lim sqrt(x)^n = sqrt(c)^n x->c

What does L mean?

Limit

Infinite(asymptotic) Discontinuity?

Means the limit approaches infinite at some point. The curve has asymptotes.

Non-removable Discontinuity

No limit -Unbounded Behavior- Asymptotes of any kind, could be going in different directions -Behavior that differ from the right and left -Oscillating behavior- like an ekg, has no significant behavior and therefore doesn't approach anything so no limit.

Do rationalizing and other methods always work?

No, and in this case just plug into the original equation and states that the limit dne

The limit of f(x) does not exist at f(c), but the other conditions are met.

Non-removable discontinuity because there is no limit

How is a limit evaluated graphically?

Observe the value ON THE CURVE (ignore the point) simultaneously approached as you approach a on the x-axis from the left and right.

Removable discontinuity?

Occurs when the limit exists, but it is not equal to f(a). This results in a hole on the curve.

What is a limit EVALUATED at infinite?

Occurs when the x-value increases/decreases without bound. The y-value is ± ∞ or a finite number. It determines the END BEHAVIOR of a function.

How is a limit evaluated NUMERICALLY?

Plug into values close to a from the right and left. Observe the trend. Usually the last resort of the three ways to evaluate a limit.

Which functions are always continuous on open intervals except for in some piecewise functions

Polynomial and trig functions

Functions with limits that can be found through Direct Substitution

Polynomials Radicals Trig Functions

Remember!

Show all work to receive full credit

When evaluating a limit analytically what should you always try first?

Simply try to solve by plugging a (what x approaches) into the function.

How to determine a removable discontinuity analytically

When rearranging an equation to simplify so that it agrees at all but one point, that one point is the removable discontinuity because it can be removed and and the graph appears the same, minus the hole where the x value was.

What can an absolute value function be rewritten as?

a piecewise lxl/x {-1 x<0, 1 x>0, x cannot = 0 then plug in for zero because that is where there is a possible discontinuity lim x->0^-(from the left)= -1 lim x-.0^+(from the right)=1 limit dne bc left and right behavior differ

First thing to do when finding limits

always try direct substitution first

Asymptotes and Discontinuity

anytime you have asymptote it is a non-removable discontinuity. however, the converse of this statement is not true.

find the constant a, or the constants a and b, such that the function is continuous on the entire real line.

first find the possible the discontinuity and plug it in as c to find the limit. then plug it into the next piece to solve for a by setting it equal to the limit obtain in the first step. this is all done bc we are told to assume that the function is continuous therefore the limits must be the same

Limits of piecewise functions and V.A.'s

increase without bound

Scalar Multiple

lim [b f(x)] = bL lim 2(x^2)= 2(4) b=2, l=f(c) x->c x->2

Sum or Difference

lim [f(x) +/- g(x)] = L+/- K L=lim of f(x) K= lim of g(x) x->c

Power

lim [f(x)]^n = L^n lim (x-1)^2 = (1)^2 L=1 n=2 x->c x->2

Product

lim [f(x)g(x)] = LK L=lim of f(x) K= lim of g(x) x->c

Quotient

lim f(x)/g(x) = L/K, provided K is not equal to 0 x->c

Special Limits

lim sinx/x =1 lim 1-cosx/x= 0 x->0 x->0 rule only applies when x's have the same coefficients lim sin3x/2x= (1/2)sin3x/x= 3/3(1/2)sin3x/3x= (3/2)sin3x/3x= (3/2)(1)= 3/2 x->0

Does f(c) matter in determining a limit?

no, the left and right behavior it what determines a limit. If the behaviors match there is a limit, so although f(c)=1 the limit can be 3 or any other real number

To receive full credit

plug for each condition. when asked to discuss continuity, state which

When asked to discuss the continuity of each function,

state the discontinuity and explain why. also, if possible, plug the equation in for the three conditions to express it analytically as well.

For rational limits: when a small number is being approached in the denominator______

the limit is equal to infinite. When a number is divided by a small number it produces a big number. So it makes sense that the limit approaches ±∞.

Continuity on an Open Interval

A function is continuous on an open interval (a,b) when the function is continuous at each point in the interval. A function that is continuous on the entire real number line (-∞,∞).

Removable Discontinuity

A hole in a graph. That is, a discontinuity that can be "repaired" by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can made connected by filling in a single point.

What is a infinite limit?

A limit that increases or decreases without bound as the limit gets closer to the x-value. Basically, the y-values get really big in either positive/negative directions as you get close. The limit is considered to not exist as infinite is not a number.

What are the three rules for a limit to be continuous?

All 3 ensure there's no discontinuities. Condition 3 is the short-hand rule to immediately tell if something is continuous. The former two conditions are just an extension of it.

Continuous function?

Basically if there is no discontinuity along an interval. That is, the graph can be drawn in 1 continuous motion without lifting your pencil.

Intermediate Value Theorem

Basically, on a continuous interval. Each value between the two endpoints have corresponding pairs between the inputs and outputs.

Fractional Power limit law?

Basically, the limit must be positive if you are rooting by an even number.

What does C mean?

C is the constant real number usually an x value that we try to determine to limit of.

For a rational limit EVALUATED AT infinite: what is true if the degree of numerator < degree of denominator?

The limit is equal to 0 since infinite is in the denominator.

For rational limits: when a big number such as ±∞ is approached in the denominator ______

The limit is equal to 0. Dividing by a big number produces a small number so it makes sense that it approaches 0.

For a rational limit EVALUATED at infinite: what is true if the degree of numerator = degree of denominator?

The limit is equal to the ratio of their leading coefficients.

For a rational limit EVALUATED at infinite: what is true if the degree of numerator > degree of denominator

The limit is ∞ or -∞

If f(c) is not defined, but the other conditions are met.

There is still a limit for when x->c so the discontinuity is removable.

The limit of f(x) exists at f(c), but is not equal to f(c), but the other conditions are met.

There is still a limit for when x->c so the discontinuity is removable.

Basic Limits

This just means you plug c in x's place to find limit in direct substitution Also if f(x)=2. y=2 so now matter what x value it will always be approaching 2 lim 2 = 2 lim x = c lim x^n = c^n x->c x->c x->c