First slide

Introduction to limits

Question

The value of $lim_{h \to 0} \frac{\ln (1 + 2 h) - 2 \ln (1 + h)}{h^{2}}$ , is

Moderate

A

1
B

-1
C

0
D

none of these

Solution

We have,
$\begin{array}{l} lim_{h \to 0} \frac{\log (1 + 2 h) - 2 \log (1 + h)}{h^{2}} \\ = lim_{h \to 0} \frac{\log \{\frac{1 + 2 h}{(1 + h)^{2}}\}}{h^{2}} = - lim_{h \to 0} \frac{\log \{\frac{(1 + h)^{2}}{1 + 2 h}\}}{h^{2}} \\ = - lim_{h \to 0} \frac{\log \{1 + \frac{h^{2}}{1 + 2 h}\}}{\{\frac{h^{2}}{1 + 2 h}\} (1 + 2 h)} = - 1 = - 1 . \end{array}$

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