# Evaluate the following limit. lim_{xrightarrow0^+}(1+5x)^{2/x}

Evaluate the following limit.
$\underset{x\to {0}^{+}}{lim}\left(1+5x{\right)}^{2/x}$
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Evaluate limits:
$\underset{x\to {0}^{+}}{lim}\left(1+5x{\right)}^{2/x}$
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:
$g\left(x\right)=\frac{2}{x}\mathrm{ln}\left(1+5x\right),f\left(u\right)={e}^{u}$
$\underset{x\to {0}^{+}}{lim}g\left(x\right)=\underset{x\to {0}^{+}}{lim}\frac{2}{x}\mathrm{ln}\left(1+5x\right)$
$=2\cdot \underset{x\to {0}^{+}}{lim}\left(\frac{\mathrm{ln}\left(1+5x\right)}{x}\right)$
$\underset{x\to {0}^{+}}{lim}\left(\frac{\mathrm{ln}\left(1+5x\right)}{x}\right)=\frac{0}{0}$
LHospital
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