$\tiny{^\text{(Don't)}}$ Use L'Hopital to solve this

Calculus Level 4

$\lim_{x\to \infty} x \left( A-x \left( B-x \left( C-\left( 1+\frac1x\right)^x \right) \right) \right)$ If the above limit exists, and it equals $\dfrac ab\times e$ , where $a$ and $b$ are coprime positive integers, then find $a+b$ .

The answer is 23.

1 solution

C Anshul
Jul 11, 2018

$Hints:$

Firstly put $t=\frac{1}{x}$ Then simplify.

Use taylor series of $(1+x)^{\frac{1}{x}}=e(1-\frac{x}{2}+\frac{11}{24}x^{2}-\frac{7}{16}x^{3}.......)$

After simplifying limit u will observe that our limit is just negative of coefficient of $x^{3}$ .

So $a=7,b=16$ .

To find taylor series: $e^{ln(1+x)/x}=e^{1-\frac{1}{2}x+\frac{1}{3}x^{2}....}=e(e^{-x/2})(e^{x^{2}/3}).....$

Now use series of $e^{-x/2}$ and $e^{x^{2}/3}$ .

Nice solution

Jitender Sharma - 2 years, 9 months ago

(Don’t) \tiny{^\text{(Don't)}} (Don’t) Use L'Hopital to solve this

The answer is 23.

1 solution

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$\tiny{^\text{(Don't)}}$ Use L'Hopital to solve this