[tex]\lim_{n \to \infty} (\frac{2x^{2}+3 }{2x^2+5})^{-3x^2} = \lim_{n \to \infty} (1+\frac{2x^{2}+3 }{2x^2+5}-1)^{-3x^2} = \lim_{n \to \infty} (1+\frac{-2 }{2x^2+5})^{-3x^2}=\lim_{n \to \infty} (1+\frac{-2 }{2x^2+5})^{\frac{2x^2+5}{-2}*\frac{-2}{2x^2+5}*(-3x^2)}=\lim_{n \to \infty}e^{\frac{-2}{2x^2+5}*(-3x^2) }=\lim_{n \to \infty}e^{\frac{6x^2}{2x^2+5} }=e^3\\\\[/tex]
[tex]\lim_{n \to \infty} (\frac{1+4x}{3+4x} )^{3x+1} = \lim_{n \to \infty} (1+\frac{1+4x}{3+4x} -1)^{3x+1} = \lim_{n \to \infty} (1+\frac{-2}{3+4x})^{3x+1}=\lim_{n \to \infty} (1+\frac{-2}{3+4x})^{\frac{3+4x}{-2}*\frac{-2}{3+4x}*(3x+1) }=\lim_{n \to \infty}e^{\frac{-6x-2}{4x+3} }=e^{\frac{-3}{2} }[/tex]
