\infty} \frac{x^3+1}{2x^3+1}=\\\\ lim_{x->\infty} \frac{1+\frac{1}{x^3}}{2+\frac{1}{x^3}}=\\\\ \frac{1+0}{2+0}=\frac{1}{2}=0.5" alt="lim_{x->\infty} \frac{x^3+1}{2x^3+1}=\\\\ lim_{x->\infty} \frac{1+\frac{1}{x^3}}{2+\frac{1}{x^3}}=\\\\ \frac{1+0}{2+0}=\frac{1}{2}=0.5" align="absmiddle" class="latex-formula">
7} \frac{\sqrt{2+x}-3}{x-7}=\\\\ lim_{x->7} \frac{\sqrt{2+x}-3}{x+2-9}=\\\\ lim_{x->7} \frac{\sqrt{2+x}-3}{\sqrt{x+2}-3)(\sqrt{x+2}+3)}=\\\\ lim_{x->7} \frac{1}{\sqrt{x+2}+3}=\\\\ \frac{1}{\sqrt{7+2}+3}=\frac{1}{6}" alt="lim_{x->7} \frac{\sqrt{2+x}-3}{x-7}=\\\\ lim_{x->7} \frac{\sqrt{2+x}-3}{x+2-9}=\\\\ lim_{x->7} \frac{\sqrt{2+x}-3}{\sqrt{x+2}-3)(\sqrt{x+2}+3)}=\\\\ lim_{x->7} \frac{1}{\sqrt{x+2}+3}=\\\\ \frac{1}{\sqrt{7+2}+3}=\frac{1}{6}" align="absmiddle" class="latex-formula">
0} \frac{arcsin (3x)}{5x}=\\\\ lim_{x->0} \frac{arcsin (3x)}{3x}*\frac{3}{5}=\\\\ 1*\frac{3}{5}=0.6" alt="lim_{x->0} \frac{arcsin (3x)}{5x}=\\\\ lim_{x->0} \frac{arcsin (3x)}{3x}*\frac{3}{5}=\\\\ 1*\frac{3}{5}=0.6" align="absmiddle" class="latex-formula">
\infty} (\frac{2x-1}{2x+2})^x=\\\\ lim_{t->\infty} (\frac{t-1}{t+2})^{\frac{t}{2}}=\\\\ lim_{t->\infty} (\frac{t+2-3}{t+2})^{\frac{t}{2}}=\\\\ lim_{t->\infty} (1-\frac{3}{t+2})^{\frac{t}{2}}=\\\\ lim_{t->\infty} (1+\frac{1}{\frac{-(t+2)}{3})}^{\frac{t}{2}}=\\\\ lim_{t->\infty} (1+\frac{1}{\frac{-(t+2)}{3}})}^{\frac{t}{2}}=\\\\ lim_{u->\infty} (1+\frac{1}{\frac{-u}{3}})}^{\frac{u-2}{2}}=\\\\ " alt="lim_{x->\infty} (\frac{2x-1}{2x+2})^x=\\\\ lim_{t->\infty} (\frac{t-1}{t+2})^{\frac{t}{2}}=\\\\ lim_{t->\infty} (\frac{t+2-3}{t+2})^{\frac{t}{2}}=\\\\ lim_{t->\infty} (1-\frac{3}{t+2})^{\frac{t}{2}}=\\\\ lim_{t->\infty} (1+\frac{1}{\frac{-(t+2)}{3})}^{\frac{t}{2}}=\\\\ lim_{t->\infty} (1+\frac{1}{\frac{-(t+2)}{3}})}^{\frac{t}{2}}=\\\\ lim_{u->\infty} (1+\frac{1}{\frac{-u}{3}})}^{\frac{u-2}{2}}=\\\\ " align="absmiddle" class="latex-formula">
\infty} (1+\frac{1}{\frac{-u}{3}})}^{\frac{u}{2}}*(1+\frac{1}{\frac{-u}{3}})=\\\\ e^{-\frac{3}{2}}*1=\frac{1}{e* \sqrt{e}}" alt="lim_{u->\infty} (1+\frac{1}{\frac{-u}{3}})}^{\frac{u}{2}}*(1+\frac{1}{\frac{-u}{3}})=\\\\ e^{-\frac{3}{2}}*1=\frac{1}{e* \sqrt{e}}" align="absmiddle" class="latex-formula">