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Maths-

General

Easy

Question

If $stack l i m with x rightwards arrow 0 below 1 plus x to the power of 2 end exponent plus fraction numerator f open parentheses x close parentheses over denominator x end fraction$ is equal to e⁷, then $stack l i m with x greater-than or slanted equal to 0 below fraction numerator f open parentheses x close parentheses over denominator x to the power of 3 end exponent end fraction equals horizontal ellipsis horizontal ellipsis$

7
4
5
6

The correct answer is: 6

$table row cell l equals stack l i m with x equals 0 below fraction numerator e to the power of 6 end exponent over denominator stack 2 with _ below end fraction plus fraction numerator x to the power of 3 end exponent plus f left parenthesis x right parenthesis over denominator x end fraction fraction numerator u to the power of j divided by x to the power of 2 end exponent end exponent over denominator x end fraction end cell row cell text end text text i end text text f end text text end text A equals fraction numerator x to the power of 3 end exponent plus f left parenthesis x right parenthesis over denominator x end fraction comma text end text text t end text text h end text text e end text text n end text text end text end cell end table$
$l equals stack l i m with x E 0 below open curly brackets left square bracket 1 plus A right square bracket to the power of logical and end exponent close curly brackets to the power of fraction numerator A over denominator x to the power of 2 end exponent end fraction end exponent equals e to the power of l i m fraction numerator A over denominator x to the power of 2 end exponent end fraction end exponent$
$stack l i m with x equals 0 below fraction numerator A over denominator x to the power of 2 end exponent end fraction equals stack l i m with x equals 0 below fraction numerator x to the power of 3 end exponent plus f open parentheses x close parentheses over denominator x to the power of 3 end exponent end fraction equals stack l i m with x equals 0 below 1 plus fraction numerator f open parentheses x close parentheses over denominator x to the power of 3 end exponent end fraction$
$table row cell text end text text n end text text o end text text w end text text , end text text end text 1 plus stack l i m with x equals 0 below fraction numerator f open parentheses x close parentheses over denominator x to the power of 3 end exponent end fraction equals 7 comma text end text text ( end text text G end text text i end text text v end text text e end text text n end text text ) end text text end text end cell row cell stack l i m with x subscript 0 end subscript 0 below fraction numerator f open parentheses x close parentheses over denominator x to the power of 3 end exponent end fraction equals 6 end cell end table$

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A particle when projected in vertical plane moves along smooth surface with initial velocity 20m/s at an angle of 60°, so that its normal reaction on the surface remains zero throughout the motion. Then the slope of the tangent to the surface at height 5 m from the point of projection will be

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physics-General

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