Question

Find the following limits 1) lim_(x->0)tan(4x)/(x+sin(2x)) 2) lim_(x->oo)(e^(2x) + x)^(1/x)

Limits and continuity
ANSWERED
asked 2021-09-07
Find the following limits
1) \(\displaystyle\lim_{{{x}\to{0}}}\frac{{\tan{{\left({4}{x}\right)}}}}{{{x}+{\sin{{\left({2}{x}\right)}}}}}\)
2) \(\displaystyle\lim_{{{x}\to\infty}}{\left({e}^{{{2}{x}}}+{x}\right)}^{{\frac{{1}}{{x}}}}\)

Expert Answers (1)

2021-09-08
1) \(\displaystyle\lim_{{{x}\to{0}}}\frac{{\tan{{\left({4}{x}\right)}}}}{{{x}+{\sin{{\left({2}{x}\right)}}}}}=\lim_{{{x}\to{0}}}\frac{{{4}{{\sec}^{{2}}{\left({4}{x}\right)}}}}{{{1}+{2}{\cos{{\left({2}{x}\right)}}}}}=\frac{{{4}{{\sec}^{{2}}{\left({0}\right)}}}}{{{1}+{2}{\cos{{\left({0}\right)}}}}}=\frac{{4}}{{3}}\)
2) \(\displaystyle\lim_{{{x}\to{0}}}{\left({e}^{{{2}{x}}}+{x}\right)}^{{\frac{{1}}{{x}}}}=\lim_{{{x}\to{0}}}{e}^{{\frac{{\log{{\left({e}^{{{2}{x}}}+{x}\right)}}}}{{x}}}}={e}^{{\lim_{{{x}\to{0}}}{\left(\frac{{\log{{\left({e}^{{{2}{x}}}+{x}\right)}}}}{{x}}\right)}}}=\)
\(\displaystyle={e}^{{\lim_{{{x}\to{0}}}{\left(\frac{{{2}{e}^{{{2}{x}}}}}{{{e}^{{{2}{x}}}+{x}}}\right)}={e}^{{\lim_{{{x}\to{0}}}{\left(\frac{{{1}+{2}{e}^{{{2}{x}}}}}{{{e}^{{{2}{x}}}+{x}}}\right)}={e}^{{3}}}}}}\)
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