Question

Вычислить пределы, не применяя правило Лопиталя.

Answer 1

$limlimits_{x to 3} frac{sqrt{2x - 1} - sqrt{5}}{5x - 15}\t = x - 3, x = t + 3\limlimits_{t to 0} frac{sqrt{2t + 5} - sqrt{5}}{t} = limlimits_{t to 0} sqrt{5}frac{sqrt{1 + frac{2t}{5}} - 1}{t} =limlimits_{t to 0} sqrt{5} frac{t}{5t} = frac{1}{sqrt{5}}$

$limlimits_{x to 0} frac{2x,tg(4x)}{1 - cos(8x)} = limlimits_{x to 0} frac{2x , tg(4x)}{2sin^2(4x)} = limlimits_{x to 0} frac{2x, (4x)}{2 (4x)^2} = limlimits_{x to 0} frac{1}{4} = frac{1}{4}$

$limlimits_{x to infty} Bigl(frac{8x - 7}{8x + 3} Bigr)^{x - 1} = limlimits_{x to infty}Bigl(1 + frac{-10}{8x + 3} Bigr)^{x - 1} = limlimits_{x to infty} e^{frac{-10x + 10}{8x + 3}} = limlimits_{x to infty} e^{-frac{10}{8}} = e^{-frac{10}{8}}$