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Chapter 5: Q.2TB (page 477)

Evaluate the limit \(\lim_{x\rightarrow \infty}x^{-\frac{4}{3}}\).

Short Answer

Expert verified

\(0\)

Step by step solution

01

Given Infromation

Consider the function \(\lim_{x\rightarrow \infty}x^{-\frac{4}{3}}\).

02

Evaluate the limit

Let, \(L=\lim_{x\rightarrow \infty}x^{-\frac{4}{3}}\).

\(L=\lim_{x\rightarrow \infty}\frac{1}{x^{\frac{4}{3}}}\).

\(L=\frac{1}{\infty}\).

\(L=0\)

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Most popular questions from this chapter

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∫1x2−4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving∫1x2−4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving∫1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving∫x2+4−5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2−a2.

(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=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.

Why don’t we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution x=atan⁡uwhen we see x2+a2, even though we can’t use the substitution x=asin⁡uunless the integrand involves the square root ofa2−x2? (Hint: Think about domains.)

Solve the integral

∫1x2+9dx

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.

2(x+2)2

Solve the integral:∫lnxdx
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