First slide

Introduction to limits

Question

If f(x) = $\lim_{n \to \infty} n (x^{\frac{1}{n}} - 1),$ then for $x > 0, y > 0, f (xy)$ is equal to

Moderate

A

$\begin{array}{l} f (x) f (y) \end{array}$
B

$f (x) + f (y)$
C

$f (x) - f (y)$
D

none of these

Solution

$\begin{array}{l} f (x) = \lim_{n \to \infty} n (x^{\frac{1}{n}} - 1) \\ = \lim_{n \to \infty} \frac{x^{\frac{1}{n}} - 1}{\frac{1}{n}} \\ = \lim_{m \to 0} \frac{x^{m} - 1}{m} (where \frac{1}{n} is replaced by m) \\ = \ln x \\ or f (xy) = lnx + \ln y = f (x) + f (y) \end{array}$

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