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Chapter 14: Q. 39 (page 883)

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

limx鈫�2x3-2x2+4x-8x2+x-6.

Short Answer

Expert verified

The answer is 85.

Step by step solution

01

Step. 1 Given Information

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

f(x)=x3-2x2+4x-8x2+x-6

Put x=2in the numerator we get,

x3-2x2+4x-8=23-2(4)+4(2)-8=8-8+8-8=0.

Put x=2in the denominator we get,

x2+x-6=4+2-6=0

Since both numerator and denominator gives 0 means they both have x=2as their common root.

02

Step. 2 Factorizing 

Numerator ,x3-2x2+4x-8=x2(x-2)+4(x-2)=(x2+4)(x-2).

Denominator,x2+x-6=x2+3x-2x-6=x(x+3)-2(x+3)=(x+3)(x-2).

So,

limx鈫�2x3-2x2+4x-8x2+x-6=limx鈫�2(x2+4)(x-2)(x+3)(x-2)=limx鈫�2(x2+4)(x+3).

Now we can put the limit directly.

03

Step. 3 Final calculation of the limit

limx鈫�2x2+4x+3=4+42+3=85.

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