04_Ex 1.4

Exercise Questions

Question	Link
Q1. Determine the left hand and right-hand limit	1.4 Q-1
Q2. Discuss the continuity of $f(x)$ at $x = c$	1.4 Q-2
Q3. If $f(x) = \begin{cases} 3x & \text{if } x \leq -2 \\ x^2 - 1 & \text{if } -2 < x < 2 \\ 3 & \text{if } x \geq 2 \end{cases}$	1.4 Q-3
Q4. If $f(x) = \begin{cases} x + 2 & \text{if } x \leq -1 \\ c + 2 & \text{if } x > -1 \end{cases}$	1.4 Q-4
Q5. Find the values $m$ and $n$ , so that $f(x)$ is continuous at $x = 3$	1.4 Q-5
Q6. If $f(x) = \begin{cases} \frac{\sqrt{2x+5} - \sqrt{x+7}}{x-2} & \text{if } x \neq 2 \\ K^r & \text{if } x = 2 \end{cases}$	1.4 Q-6

Overview

This Exercise introduces the fundamental concepts of limits and continuity, which are foundational to understanding calculus. The study of limits helps in analyzing the behavior of functions as they approach specific points, while continuity ensures that a function behaves predictably and without sudden jumps or breaks. These concepts have vast applications in understanding real-world phenomena, such as motion, optimization, and instantaneous rates of change.

Key Concepts

Limit of a Function

A limit represents the value that a function approaches as the input approaches a certain point. It is critical in defining derivatives and integrals. The limit of a function at a point $x = c$ is expressed as: $\lim_{x \to c} f(x)$

Continuity of a Function

A function is continuous at $x = c$ if the following conditions are met:

$f(c)$ is defined.
$\lim_{x \to c} f(x)$ exists.
$\lim_{x \to c} f(x) = f(c)$

Left-hand and Right-hand Limits

These are the limits of a function as the variable approaches a point from the left (denoted as $L.H.L$ ) or from the right (denoted as $R.H.L$ ).

Important Formulas

Limit:
The limit of a function $f(x)$ as $x \to c$ is defined as: $\lim_{x \to c} f(x) = L$
where $L$ is the value that $f(x)$ approaches as $x \to c$ .
Continuity:
A function $f(x)$ is continuous at $x = c$ if:
$\lim_{x \to c} f(x) = f(c)$

Tips and Tricks

When calculating limits, always check if direct substitution results in an indeterminate form such as $\frac{0}{0}$ . In such cases, try factoring, rationalizing,.
A function with a jump, infinite, or removable discontinuity will not be continuous at that point.
For R.H.L, L.H.L functions, check the limits from both sides to determine continuity.

Summary

This chapter covered the concepts of limits and continuity, emphasizing their significance in calculus. We examined how to calculate left-hand and right-hand limits, as well as how to determine if a function is continuous at a particular point. Understanding these concepts is essential for studying more advanced topics like derivatives and integrals.

Reference

By Sir Shazad Sair: