List of top Engineering Mathematics Questions on Limits

Evaluate the following limit: \[ \lim_{x \to 0} \frac{\ln(1+x)}{2 \sin(x)} \quad {(rounded off to two decimal places).} \]

Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]

Evaluate the following limit:
\[ \lim_{x \to 0} \frac{\ln(1+x)}{2 \sin(x)} \quad \text{(rounded off to two decimal places).} \]

Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]

Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]

Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]

Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]

Consider the sequence \[ x_n = 0.5x_{n-1} + 1, n = 1, 2, \ldots \text{ with } x_0 = 0. \text{ Then } \lim_{n \to \infty} x_n \text{ is} \]