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Compute the limit of the ratio of functions satisfying given equations

by RoRi

Define two functions as follows:

    \[ g(x) = xe^{x^2}, \qquad f(x) = \int_1^x g(t) \left( t + \frac{1}{t} \right) \, dt. \]

Using these definitions find the limit as x \to +\infty of the ratio of the second derivatives:

    \[ \lim_{x \to +\infty} \frac{f''(x)}{g''(x)}. \]

First, we take the derivatives of the given functions,

    \begin{align*} &&f(x) &= \int_1^x g(t) \left( t + \frac{1}{t} \right) \, dt \\[9pt] \implies &&f'(x) &= g(x) \left( x + \frac{1}{x} \right) \\[9pt] &&&= xe^{x^2} \left( x + \frac{1}{x} \right) \\[9pt] &&&= x^2 e^{x^2} + e^{x^2} \\[9pt] \implies && f''(x) &= 2x^3 e^{x^2} + 2xe^{x^2} + 2x e^{x^2} \\[9pt] &&&= (2x^3 + 4x) e^{x^2}. \end{align*}

And for g(x),

    \begin{align*} &&g(x) &= xe^{x^2} \\[9pt] \implies && g'(x) &= 2x^2 e^{x^2} + e^{x^2} \\[9pt] \implies && g''(x) &= 4x^3 e^{x^2} + 4xe^{x^2} + 2xe^{x^2} \\[9pt] &&& = (4x^3 + 6x) e^{x^2}. \end{align*}

Now, we evaluate the requested limit,

    \begin{align*} \lim_{x \to +\infty} \frac{f''(x)}{g''(x)} &= \lim_{x \to +\infty} \frac{(2x^3+4x)e^{x^2}}{(4x^3 + 6x)e^{x^2}} \\[9pt] &= \lim_{x \to +\infty} \frac{2 + \frac{4}{x^2}}{4 + \frac{6}{x^2}} \\[9pt] &= \frac{1}{2}. \end{align*}

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