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Chapter 5: Q.1TB (page 477)

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

Short Answer

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$0$

Step by step solution

Given Information

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

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 $\int x {(x^{2} - 1)}^{2} d x$ .

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

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

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

$\int \frac{x^{2} + 9}{x^{\frac{3}{2}}} d x$

Solve given definite integral.

$\int_{- 1}^{1} x^{3} \sqrt{9 x^{2} - 1} d x$

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 solving $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(f) True or False: Trigonometric substitution doesn’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 $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

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

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

$x^{2} - 4 x - 8$

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

Short Answer

Step by step solution

Given Information

Evaluate the limit

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Most popular questions from this chapter

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