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Chapter 14: Q. 38 (page 883)

In Problems 7鈥� 42, find each limit algebraically.

limx鈫�-1x3+x2+3x+3x4+x3+2x+2.

Short Answer

Expert verified

The answer is 4.

Step by step solution

01

Step. 1 Given Information

Firstly, we check whether the given function is in indeterminant form or not.

f(x)=x3+x2+3x+3x4+x3+2x+2

Put x=-1in the numerator we get,

x3+x2+3x+3=(-1)3+(-1)2+3(-1)+3=-1+1-3+3=0.

Put x=-1in the denominator we get,

x4+x3+2x+2=(-1)4+(-1)3+2(-1)+2=1-1-2+2=0.

Since both numerator and denominator gives 0 means they both have x=-1as their common root.

02

Step. 2 Factorizing

Denominator,x4+x3+2x+2=x3(x+1)+2(x+1)=(x3+2)(x+1).

Numerator ,x3+x2+3x+3=x2(x+1)+3(x+1)=(x2+3)(x+1).

So,

limx鈫�-1x3+x2+3x+3x4+x3+2x+2=limx鈫�-1(x2+3)(x+1)(x3+2)(x+1)=limx鈫�-1x2+3x3+2.

Now we can put the limit directly.

03

Step. 3 Final calculation of the limit 

limx鈫�-1x2+3x3+2=1+3-1+2=41=4.

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