Calculas

If limx→a− f(x)=L and limx→a+ f(x)=M​, where L and M are finite real​ numbers, then what must be true about L and M in order for limx→a f(x) to​ exist?

L=M

Which of the following is the correct formula for average​ velocity?

vav= s(b)-s(a)/b-a

What is a vertical​ asymptote?

A vertical asymptote is a vertical line x=​a, which the graph of a function approaches as x approaches a

When limx→a f(x) exists, it always equals​ f(a). State whether this statement is true or false.

False. The limit​ (if it​ exists) depends on values of f near​ a, but it does not depend on the value of​ f(a)

State the precise definition of lim z→c h(z)=P.

For any number ε>0​, there is a corresponding number δ>0 such that h(z)−P<ε whenever 0<z−c<δ

We informally describe a function f to be continuous at a if its graph contains no holes or breaks at a. Explain why this is not an adequate definition of continuity.

A graph may not be precise enough to show points of discontinuity in a complicated function

Explain the meaning of limx→b− r(x)=K.

As x approaches b from the left​, the value of r​(x) approaches K

Explain the meaning of limx→−∞ f(x)=10

As x becomes arbitrarily large in magnitude and​ negative, the value of​ f(x) approaches 10

State whether each of the following functions are continuous or not for all values in their domain: T(t)= temperature t minutes after midnight in Chicago on January 1.

The function​ T(t) is continuous because the temperature changes gradually as time​ increases, with no jumps in between

State whether each of the following functions are continuous or not for all values in their domain: a(t)= altitude of a skydiver t seconds after jumping from a plane.

The function​ a(t) is continuous because the​ skydiver's altitude decreases as time​ increases, with no breaks in between

State whether each of the following functions are continuous or not for all values in their domain: ​n(t)= number of quarters needed to park in a metered parking space for t minutes.

The function​ n(t) is discontinuous because the number of quarters must be a whole number

State whether each of the following functions are continuous or not for all values in their domain: p(t)= number of points scored by a baseball player after t minutes of a baseball game.

The function​ p(t) is discontinuous because the number of points must be a whole number

Determine whether the following statements are true and give an explanation or counterexample: For a given ε>​0, there is one value of δ>0 for which f(x)−L<ε whenever 0<x−a<δ.

The statement is false. For any δ>​0, every δ0 with 0<δ0<δ also satisfies the condition that f(x)−L<ε because 0<x−a<δ holds whenever 0<x−a<δ0 holds

Determine whether the following statements are true and give an explanation or counterexample: The limit limx→a f(x)=L means that given an arbitrary δ>​0, an ε>0 can always be found such that f(x)−L<ε whenever 0<x−a<δ.

The statement is false. It may not be possible to bound the function to within ε of L on an interval of x within δ of a for arbitrary δ​, even if the limit exists

Determine whether the following statements are true and give an explanation or counterexample: The limit lim x→a f(x)=L means that given any arbitrary ε>​0, a δ>0 can always be found such that f(x)−L<ε whenever 0<x−a<δ.

The statement is true. This is the precise definition of the limit

Determine whether the following statements are true and give an explanation or counterexample: If x−a<δ​, then a−δ<x<a+δ.

The statement is​ true, because x−a<δ means −δ<x−a<δ by the definition of the absolute​ value, from which the above follows

For what values of a does limx→a r(x)=r(a) if r is a rational​ function?

Those values of a for which the denominator of the function r is not zero

Give the three conditions that must be satisfied by a function to be continuous at a point.

lim x→a f(x) exists, f(a) is defined, lim x→a f(x)=f(a)

How is limx→a p(x) calculated if p is a polynomial​ function?

lim x→a p(x)= ​p(a)

A function is continuous from the right at a if

lim x→a+ f(x)= f(a)

A function is continuous from the left at a if

lim x→a- f(x)= f(a)

How are limx→a− p(x) and limx→a+ p(x) calculated if p is a polynomial​ function?

lim x→a- p(x)= lim x→a+ p(x)= p(a)

Which of the following is the correct formula for the slope of the secant​ line?

msec= s(b)-s(a)/b-a

Which of the following is the correct formula for the slope of the tangent​ line?

mtan= lim t→a f(t)-f(a)/t-a