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Question Number 78325 by jagoll last updated on 16/Jan/20

$$$$$$\underset{x\to \mathrm{\infty }}{\lim }{\left(\frac{5x^2+3x}{5x^2-2x}\right)}^{3x-1}$$

Commented by john santu last updated on 16/Jan/20

$$consider\underset{x\to \mathrm{\infty }}{\lim }{(1+\frac{1}{\left(\frac{5x^2-2x}{5x}\right)})}^{\frac{5x^2-2x}{5x}}=e$$

Commented by john santu last updated on 16/Jan/20

$$byshortcut$$$$e^{\underset{x\to \mathrm{\infty }}{\lim }(3x-1)\left(\frac{5x}{5x^2-2x}\right)}=e^3$$

Commented by mathmax by abdo last updated on 16/Jan/20

$$letA\left(x\right)={\left(\frac{5x^2+3x}{5x^2-2x}\right)}^{3x-1}\Rightarrow A\left(x\right)=e^{(3x-1)ln\left(\frac{5x^2+3x}{5x^2-2x}\right)}$$$$wehaveln\left(\frac{5x^2+3x}{5x^2-2x}\right)=ln\left(\frac{5x^2-2x+5x}{5x^2-2x}\right)=ln(1+\frac{5x}{5x^2-2x})$$$$\sim \frac{5x}{5x^2-2x}\sim \frac{1}{x}(x\to +\mathrm{\infty })\Rightarrow A\left(x\right)\sim e^{\frac{3x-1}{x}}=e^3.e^{-\frac{1}{x}}\Rightarrow$$$${lim}_{x\to +\mathrm{\infty }}A\left(x\right)=e^3$$

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