Calculus S.M Yusuf Chapter 1 Ex 1.2 Solved Notes & Quiz
What are the Core Formulas and Definitions for this Exercise?
In BS and ADP University Mathematics, Chapter 1 Exercise 1.2 focuses on evaluating algebraic, trigonometric, and transcendental limits, as well as testing the continuity of functions at specific points. An indeterminate form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ requires algebraic manipulation (factorization, rationalization, or trigonometric identity application) prior to evaluating the limit.
Definition of Continuity: A function $f(x)$ is continuous at a point $x = c$ in its domain if and only if three conditions are satisfied simultaneously:
- $f(c)$ is well-defined (exists as a real number).
- $\lim_{x \to c} f(x)$ exists, which implies $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$.
- $\lim_{x \to c} f(x) = f(c)$.
| Limit / Continuity Theorem | Mathematical Formula | Application / Conditions |
|---|---|---|
| Squeeze Theorem Limit | $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ | $\theta$ must be measured in radians |
| Cosine Difference Limit | $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ | Trigonometric indeterminate evaluation |
| Euler's Exponential Limit (Type 1) | $\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$ | Base approaches $1$, power approaches $\infty$ |
| Euler's Exponential Limit (Type 2) | $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$ | Asymptotic evaluation as variable approaches infinity |
| Exponential Base Limit | $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$ | For any constant $a > 0, a \neq 1$ |
How to Solve All Exercise Questions Step-by-Step?
Question 1 (Part i): Evaluate the algebraic limit: $\lim_{x \to 2} \frac{x^3 - 8}{
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