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Maths-

General

Easy

Question

$L t subscript left parenthesis x rightwards arrow 0 right parenthesis left parenthesis left parenthesis 1 minus e to the power of x right parenthesis s i n invisible function application x right parenthesis divided by left parenthesis x squared plus x cubed right parenthesis equals$

1
0
-1
$e^{2}$

hint Hint:

In this question we will us the standard limits $limit as x minus 0 of fraction numerator left parenthesis e to the power of x minus 1 right parenthesis over denominator x end fraction equals 1$ and $limit as x minus greater than 0 of fraction numerator sin open parentheses x close parentheses over denominator x end fraction equals 1$ to find the limit

The correct answer is: -1

In this question we have to find limit of $limit as x minus greater than 0 of fraction numerator left parenthesis 1 minus e to the power of x right parenthesis sin open parentheses x close parentheses over denominator x squared plus x cubed end fraction$
Step1: Rearranging the expression
$limit as x minus greater than 0 of fraction numerator left parenthesis 1 minus e to the power of x right parenthesis over denominator x end fraction cross times open parentheses fraction numerator sin open parentheses x close parentheses over denominator x end fraction close parentheses open parentheses limit as x minus greater than 0 of fraction numerator 1 over denominator 1 plus x end fraction close parentheses$
Step2: Using Standard Limits
We know that $limit as x minus 0 of fraction numerator left parenthesis e to the power of x minus 1 right parenthesis over denominator x end fraction equals 1$ and $limit as x minus greater than 0 of fraction numerator sin open parentheses x close parentheses over denominator x end fraction equals 1$
=> $negative 1 cross times open parentheses limit as x minus greater than 0 of fraction numerator 1 over denominator 1 plus x end fraction close parentheses$
=> $negative 1$ .
So, the value of the limit is $negative 1$

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parallel

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