I want to find the limit:

$\sqrt{{n}^{2}+n}-\sqrt[3]{{n}^{3}-n}$

$\sqrt{{n}^{2}+n}-\sqrt[3]{{n}^{3}-n}$

Alissa Hutchinson
2022-05-12
Answered

I want to find the limit:

$\sqrt{{n}^{2}+n}-\sqrt[3]{{n}^{3}-n}$

$\sqrt{{n}^{2}+n}-\sqrt[3]{{n}^{3}-n}$

You can still ask an expert for help

Mathias Patrick

Answered 2022-05-13
Author has **22** answers

Both parts of the expression are "about n" for large n. So, you can separate the problem into finding

$\underset{n\to \mathrm{\infty}}{lim}\sqrt{{n}^{2}+n}-n$

and

$\underset{n\to \mathrm{\infty}}{lim}\sqrt[3]{{n}^{3}-n}-n,$

both of which are amenable to the relationships you wish to use.

$\underset{n\to \mathrm{\infty}}{lim}\sqrt{{n}^{2}+n}-n$

and

$\underset{n\to \mathrm{\infty}}{lim}\sqrt[3]{{n}^{3}-n}-n,$

both of which are amenable to the relationships you wish to use.

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The figure shows the surface created when the cylinder ${y}^{2}+{Z}^{2}=1$ intersects the cylinder ${x}^{2}+{Z}^{2}=1$. Find the area of this surface.

The figure is something like:

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Textbooks that use notation with explicit argument variable in the upper bound ${\int}^{x}$ for "indefinite integrals."

I dare to ask a question similar to a closed one but more precise.

Are there any established textbooks or other serious published work that use ${\int}^{x}$ notation instead of $\int $ for the so-called "indefinite integrals"?

(I believe I've seen it already somewhere, probably in the Internet, but I cannot find it now.)

So, I am looking for texts where the indefinite integral of cos would be written something like:

${\int}^{x}\mathrm{cos}(t)dt=\mathrm{sin}(x)-C$

or ${\int}^{x}\mathrm{cos}(x)dx=\mathrm{sin}(x)+C.$

(This notation looks more sensible and consistent with the one for definite integrals than the common one with bare $\int $.)

IMO, the indefinite integral of f on a given interval I of definition of f should not be defined as the set of antiderivatives of f on I but as the set of all functions F of the form

$F(x)={\int}_{a}^{x}f(t)dt+C,\phantom{\rule{2em}{0ex}}x\in I,$,

with $a\in I$ and C a constant (or as a certain indefinite particular function of such form). In other words, I think that indefinite integrals should be defined in terms of definite integrals and not in terms of antiderivatives. (After all, the integral sign historically stood for a sum.)

In this case, the fact that the indefinite integral of a continuous function f on an interval I coincides with the set of antiderivatives of f on I is the contents of the first and the second fundamental theorems of calculus:

1. The first fundamental theorem of calculus says that every representative of the indefinite integral of f on I is an antiderivative of f on I, and

2. The second fundamental theorem of calculus says that every antiderivative of f on I is a representative of the indefinite integral of f on I (it is an easy corollary of the first one together with the mean value theorem).

I dare to ask a question similar to a closed one but more precise.

Are there any established textbooks or other serious published work that use ${\int}^{x}$ notation instead of $\int $ for the so-called "indefinite integrals"?

(I believe I've seen it already somewhere, probably in the Internet, but I cannot find it now.)

So, I am looking for texts where the indefinite integral of cos would be written something like:

${\int}^{x}\mathrm{cos}(t)dt=\mathrm{sin}(x)-C$

or ${\int}^{x}\mathrm{cos}(x)dx=\mathrm{sin}(x)+C.$

(This notation looks more sensible and consistent with the one for definite integrals than the common one with bare $\int $.)

IMO, the indefinite integral of f on a given interval I of definition of f should not be defined as the set of antiderivatives of f on I but as the set of all functions F of the form

$F(x)={\int}_{a}^{x}f(t)dt+C,\phantom{\rule{2em}{0ex}}x\in I,$,

with $a\in I$ and C a constant (or as a certain indefinite particular function of such form). In other words, I think that indefinite integrals should be defined in terms of definite integrals and not in terms of antiderivatives. (After all, the integral sign historically stood for a sum.)

In this case, the fact that the indefinite integral of a continuous function f on an interval I coincides with the set of antiderivatives of f on I is the contents of the first and the second fundamental theorems of calculus:

1. The first fundamental theorem of calculus says that every representative of the indefinite integral of f on I is an antiderivative of f on I, and

2. The second fundamental theorem of calculus says that every antiderivative of f on I is a representative of the indefinite integral of f on I (it is an easy corollary of the first one together with the mean value theorem).

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For $f\left(x\right)=-\frac{x}{3}$ what is the equation of the tangent line at x=-3?

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Antiderivatives: A car is traveling at 50mi/h when the brakes are fully applied, producing a constant deceleration of 38ft/s2. What is the distance covered before the car comes to a stop?

Since the car is decelerating, our value will be negative.

${v}^{\prime}(t)=a(t)=-38$ so $v(t)=-38x=s(t)-38{x}^{2}/(2)$.

After this point I am not sure what I am supposed to do?

Since the car is decelerating, our value will be negative.

${v}^{\prime}(t)=a(t)=-38$ so $v(t)=-38x=s(t)-38{x}^{2}/(2)$.

After this point I am not sure what I am supposed to do?

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Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{t\to 2}{lim}\frac{3{t}^{2}-7t+2}{2-t}$

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Why is $\underset{s\to 0}{lim}{\int}_{0}^{\mathrm{\infty}}\frac{\mathrm{sin}(t)}{t}\cdot {e}^{-st}dt={\int}_{0}^{\mathrm{\infty}}\frac{\mathrm{sin}(t)}{t}dt$ legitimate?

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Find the first partial derivatives of the following functions.

$g(x,y,z)=2{x}^{2}y-3x{z}^{4}+10{y}^{2}{z}^{2}$