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Solve the limit \(\lim_{x\rightarrow \infty}x^{-\frac{2}{3}}\).

Short Answer

Expert verified

\(0\)

Step by step solution

01

Given Information

Consider the limits \(\lim_{x\rightarrow \infty}x^{-\frac{2}{3}}\).

02

Evaluate the limit

Let, \(L=\lim_{x\rightarrow \infty}x^{-\frac{2}{3}}\)

\(L=\lim_{x\rightarrow \infty}\frac{1}{x^{\frac{2}{3}}}\)

\(L=\lim_{x\rightarrow \infty} \frac{1}{\infty}\)

\(L=0\)

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Most popular questions from this chapter

Consider the integral โˆซx(x2โˆ’1)2dx.

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.

Solve each of the integrals in Exercises 39โ€“74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

โˆซx2+9x32dx

Solve given definite integral.

โˆซโˆ’11โ€Šx39x2โˆ’1dx

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solvingโˆซ1x2โˆ’4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solvingโˆซ1x2โˆ’4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solvingโˆซ1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingโˆซx2+4โˆ’5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2โˆ’a2.

(f) True or False: Trigonometric substitution doesnโ€™t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Complete the square for each quadratic in Exercises 28โ€“33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

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