A very nice Limit

Calculus Level 3

Given that

$\large L = \lim_{x \to \ 0} \frac{1}{x} \ln\left( \frac{a^{x} + b^{x} + c^{x}}{3}\right)$

Evalute $L$ for $abc = 24$ , where $a,b,c\in \mathbb R^{+}$

1 solution

Felix Schwarzfischer
Dec 9, 2017

By rewriting the fraction you will notice that it is the formal definition of the derivative at the point $x = 0$

$\lim_{x \to \ 0} \frac{ln(a^{x} + b^{x} + c^{x}) - ln(3)}{x - 0}$ which is the same as $\frac{d}{dx} ln(a^{x} + b^{x} + c^{x})$ at $x = 0$

Differentiate and obtain the result $\frac{ln(abc)}{3}$ . By Substituting you'll get 1.059