 # Evaluate the following limits. If you use l'Hospital's Rule, be sure to indicate puntgewelb5 2021-11-21 Answered
Evaluate the following limits. If you use lHospitals Rule, be sure to indicate when yyou are using it, and why it applies.
a) $\underset{x\to \mathrm{\infty }}{lim}\left(3\cdot {2}^{1-x}+{x}^{2}\cdot {2}^{1-x}\right)$
b) $\underset{x\to {0}^{+}}{lim}{\left(1+5x\right)}^{\frac{2}{x}}$
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Evaluate limits:
$\underset{x\to \mathrm{\infty }}{lim}\left(3\cdot {2}^{1-x}+{x}^{2}\cdot {2}^{1-x}\right)$
$\underset{x\to a}{lim}\left[f\left(x\right)±g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)±\underset{x\to a}{lim}g\left(x\right)$
$\underset{x\to \mathrm{\infty }}{lim}\left(3\cdot {2}^{1-x}+{x}^{2}\cdot {2}^{1-x}\right)=\underset{x\to \mathrm{\infty }}{lim}\left(3\cdot {2}^{1-x}\right)+\underset{x\to \mathrm{\infty }}{lim}\left({x}^{2}\cdot {2}^{-x}\right)$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}\left({2}^{1-x}\right)+2\cdot \underset{x\to \mathrm{\infty }}{lim}\left({x}^{2}\frac{1}{{2}^{x}}\right)$
Take $3\cdot \underset{x\to \mathrm{\infty }}{lim}\left({2}^{1-x}\right)$
Apply exponent rule ${a}^{x}={e}^{\mathrm{ln}\left({a}^{x}\right)}={e}^{x\mathrm{ln}\left(a\right)}$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}\left({e}^{\left(1-x\right)\mathrm{ln}\left(2\right)}\right)\right)$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}\left({2}^{1-x}\right)$
$=3\cdot \underset{x\to -\mathrm{\infty }}{lim}\left({e}^{x\mathrm{ln}\left(2\right)}\right)$
Apply the Limit Chain Rule:
$g\left(x\right)=x\mathrm{ln}\left(2\right),f\left(u\right)={e}^{u}$
$=-\mathrm{\infty }\cdot \mathrm{ln}\left(2\right)$
$=-\mathrm{\infty }$
$3\cdot \underset{x\to -\mathrm{\infty }}{lim}\left({e}^{x\mathrm{ln}\left(2\right)}\right)=3\cdot {e}^{-\mathrm{\infty }}$
$=3\cdot 0$
$=0$
Take $2\cdot \underset{x\to \mathrm{\infty }}{lim}\left({x}^{2}\frac{1}{{2}^{x}}\right)$
If $\sum {a}_{n}$ converges, then $\underset{n\to \mathrm{\infty }}{lim}\left({a}_{n}\right)=0$
Apply ratio test and check weather series is convergent or divergent.
If $|\frac{{a}_{n+1}}{{a}_{n}}|\le q$ eventually for some 0<q<1, the
###### Not exactly what you’re looking for? Lauren Fuller
b) Evaluate limits:
$\underset{x\to {0}^{+}}{lim}{\left(1+5x\right)}^{\frac{2}{x}}$
Apply exponent rule:
${a}^{x}={e}^{\mathrm{ln}\left({a}^{x}\right)}={e}^{x\cdot \mathrm{ln}\left(a\right)}$
${\left(1+5x\right)}^{\frac{2}{x}}={e}^{\frac{2}{x}\mathrm{ln}\left(1+5x\right)}$
Apply the limit Chain Rule:
$g\left(x\right)=\frac{2}{x}\mathrm{ln}\left(1+5x\right),f\left(u\right)={e}^{u}$
$\underset{x\to {0}^{+}}{lim}g\left(x\right)=\underset{x\to {0}^{+}}{lim}\frac{2}{x}\mathrm{ln}\left(1+5x\right)$
$=2\cdot \underset{x\to {0}^{+}}{lim}\left(\frac{\mathrm{ln}\left(1+5x\right)}{x}\right)$
$\underset{x\to {0}^{+}}{lim}\left(\frac{\mathrm{ln}\left(1+5x\right)}{x}\right)=\frac{0}{0}$
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