Find out the answer to "Evaluate the following limit. limx→0+(1+5x)2/x"

nagasenaz

Answered question

2020-11-30

Evaluate the following limit.

lim_{x \to 0^{+}} (1 + 5 x)^{2 / x}

okomgcae

Skilled2020-12-01Added 93 answers

Evaluate limits:

lim_{x \to 0^{+}} (1 + 5 x)^{2 / x}

Apply exponent rule:

a^{x} = e^{\ln (a^{x})} = e^{x \cdot \ln (a)}

(1 + 5 x)^{\frac{2}{x}} = e^{\frac{2}{x} \ln (1 + 5 x)}

Apply the Limit Chain Rule:

g (x) = \frac{2}{x} \ln (1 + 5 x), f (u) = e^{u}

lim_{x \to 0^{+}} g (x) = lim_{x \to 0^{+}} \frac{2}{x} \ln (1 + 5 x)

= 2 \cdot lim_{x \to 0^{+}} (\frac{\ln (1 + 5 x)}{x})

lim_{x \to 0^{+}} (\frac{\ln (1 + 5 x)}{x}) = \frac{0}{0}

LHospital

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