Consider the vectors and in . Let denote the angle between and . Find the value of as .

If is the angle between and then we have

Computing each of these pieces we have

Therefore,

Hence, as which implies as .

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Stumbling Robot

A Fraction of a Dot
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Tag: Limits

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Find the angle between the vectors *(2,4,6, …, 2n)* and *(1,3,5,…,2n-1)* as *n* goes to infinity

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Find the angle between the vectors *(1,1, …, 1)* and *(1,2,…,n)* as *n* goes to infinity

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Show that the limit and derivative cannot be interchanged when *f*_{n}(x) = (sin (nx)) / n

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Show that we cannot interchange a limit and integral for *f*_{n}(x) = nxe^{-nx2}

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Prove or disprove: *∫ f(x)* converges implies *lim f(x) = 0*

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Prove or disprove a statement relating the derivative of a function to an improper integral of the function

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Prove or disprove: If *f* is positive and *lim I*_{n} = A, then *∫ f(x)* converges to *A*

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Prove or disprove: If *lim f(x) = 0* and *lim I*_{n} = A then *∫ f(x)* converges to *A*

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

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Prove or disprove: If *f* is monotonic decreasing and *lim I*_{n} exists then *∫ f(x)* converges

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

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Prove some properties of improper integrals involving *1/x* and *sin x*

Consider the vectors and in . Let denote the angle between and . Find the value of as .

If is the angle between and then we have

Computing each of these pieces we have

Therefore,

Hence, as which implies as .

Consider the vectors and in . Let denote the angle between and . Find the value of as .

If is the angle between and then we have

Thus,

Therefore, as .

For each positive integer and and all real define

Prove that

*Proof.* First, we have

for all . Hence,

On the other hand,

Hence,

For each positive integer define

Prove that the following limit and integral cannot be interchange:

*Proof.* First, we have

On the other hand,

Hence,

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

The convergence of the improper integral

implies

*Counterexample.* The idea of the construction is a function which has rapidly diminishing area, but has a height that is not going to 0. (So, for an idea consider triangles on the real line all with height 1, but for which the base is becoming small rapidly.) To make this concrete, define

for each positive integer . Then for the improper integral we have

which we know converges. On the other hand

for all positive integers . Hence,

(since it does not exist). Hence, the statement is false.

(**Note:** For more on this see this question on Math.SE.)

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

Assume exists for all and is bounded,

for some constant for all . Then,

**Incomplete.**

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

If is positive and if

then

**Incomplete.**

Assume

Then

**Incomplete.**

If is monotonically decreasing and if

exists, then the improper integral

converges.

**Incomplete.**

- Prove the following limit formulas:
- Determine whether the following improper integrals converge:

**Incomplete.**