SEASON 1 EP 1 -- LIMITS

 S1 EPISODE 1 

LIMITS

INTRODUCTION:-

The concept of limits is a fundamental building block of calculus and mathematical analysis. It is used to define continuity, derivatives, and integrals.

In Mathematics, a limit is defined as a value that a function approaches the output for the given input values. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns about the behaviour of the function at a particular point.

Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals , the upper limit and lower limits are defined properly. Whereas in indefinite the integrals are expressed without limits, and it will have an arbitrary constant while integrating the function.

DEFINITION:-

A limit describes the value that a function approaches as the input (or variable) approaches a certain point. It is written as:

$$$$lim⁡x→cf(x)=L

This means that as $x$ gets arbitrarily close to $c$, the function $f(x)$ approaches the value $L$.

If $x$ gets close to a point $c$ but does not necessarily equal $c$, and $f(x)$ gets close to some value $L$, then $L$ is the limit of $f(x)$as $x$ approaches $c$.

TYPES OF LIMITS:-

Finite Limits at a Point

$$\lim_{x \to c} f(x) = L$$

This describes the behavior of $f(x)$ near $x = c$.

Infinite Limits

$$\lim_{x \to c} f(x) = \infty \quad \text{or} \quad \lim_{x \to c} f(x) = -\infty$$

This means that $f(x)$ increases or decreases without bound as $x$ approaches $c$.

Limits at Infinity

$$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$$

This represents the value $f(x)$ approaches as $x$ becomes arbitrarily large or negative.

 Formal Definition (ε-δ Definition of a Limit):-

For a function $f(x)$, the limit as $x$ approaches $c$ is $L$ if, for every $ϵ>\mathrm{0,\; there\; exists\; a }$$\delta > 0$ such that:

$$0 < |x - c| < \delta \implies |f(x) - L| < \epsilon$$

Explanation

• $\epsilon$ represents how close $f(x)$ is to $L$.
• $\delta$ represents how close $x$ is to $c$.
• This definition ensures that $f(x)$can be made arbitrarily close to $L$ by choosing $x$ sufficiently close to $c$.

Techniques for Finding Limits:-

Direct Substitution

Substitute $x = c$ into $f(x)$. If the function is continuous at $c$, this gives the limit.

Factoring

Factorize the expression to simplify it and remove indeterminate forms like $\frac{0}{0}$.

Rationalization

Multiply by the conjugate to simplify square root expressions.

L'Hopital's Rule

If a limit results in an indeterminate form ($\frac{0}{0}$ or $\frac{\infty}{\infty}$), use derivatives:

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$

Continuity and Limits:-

A function $f(x)$ is continuous at $x = c$ if:

1. $f(c)$ is defined,
2. $\lim_{x \to c} f(x)$
   
3. $$lim⁡x→cf(x)=f(c)

LIMIT LAWS:-

 Applications of Limits:-

1. Derivatives: Defined as the limit of the difference quotient. $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
2. Integration: Defined using limits of Riemann sums.
3. Series and Sequences: Limits determine convergence.

That's it for this episode , hope you gained knowledge and this blog was helpful for you , do share it with your friends , see you in next episode with a new topic in this series 😊.

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REFRENCES:-

• BYJUS
• KHAN ACADEMY
• WIKIPEDIA
• BOOK- Differential calculus for beginners by Joseph Edwards
• GOOGLE