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Chapter 5: Q. 65 (page 479)

The Limit Comparison Test: If two functions f(x) and g(x) behave in a similar way on [a,鈭�), then their improper integrals on [a,鈭�)should converge or diverge together. More specifically, the limit comparison test says that if

limx鈫�鈭�f(x)g(x)=L

for some positive real number L, then the improper integrals鈭�a鈭�f(x)dxand鈭�a鈭�g(x)dx either both converge or both diverge. We will learn more about limit comparison tests in Chapter 7. Use this test to determine the convergence or divergence of the integrals in Exercises 65鈥�70.

鈭�2鈭�1(x-1)2dx

Short Answer

Expert verified

The integral converges.

Step by step solution

01

Step 1. Given information.

The given integral is鈭�2鈭�1(x-1)2dx.

02

Step 2. Conclusion.

Let,

f(x)=1x2andg(x)=1(x-1)2.

Now,

limx鈫�鈭�f(x)g(x)=limx鈫�鈭�1x21(x-1)2=limx鈫�鈭�x-1x2=limx鈫�鈭�1-1x2=1Therefore,itconverges.

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