BCA Semester 2 – Mathematics II – Unit 1: Limits and Continuity (6Hrs)

$\textbf{Continuity at } x = 0$ For a function to be continuous at \( x = a \), the following three conditions must be satisfied: 1. $\text{The value } f(a) \text{ exists}$ 2. $\text{The limit } \lim{

We are asked to evaluate: \[ L = \lim{x \to \theta} \frac{x\cos\theta - \theta\cos x}{x - \theta} \] --- $\textbf{Step 1: Recognize indeterminate form}$ As \( x \to \theta \): \[ x\cos\theta - \theta

Indeterminate Forms: In calculus, an expression obtained while evaluating a limit is said to be an indeterminate form if the limit cannot be directly determined from the form alone. These forms arise

Step 1: Expand \(\tan x\) and \(\sin x\) using Maclaurin series \[ \sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \dots \] \[ \tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \dots \] --- Step 2: Compu