What is Limits?

 What is Limit?

Limit is a kind of function that describes how function would behave as the input would get as close to a value or output. Instead of plotting at the exact point, limits examine what will happen to a function as the input 'x' approaches to a value. The notation for a limit is $\lim_{x \to a}$ f(x) = L , which means that as x gets arbitrarily close to c (but not necessarily equal to a), the value of f(x) gets arbitrarily close to L. Limit is the essential topics which will be implemented in derivatives, continuity,discontinuity and even integration.

Difference between Limit and Function:

A function is a rule that assigns each input value (from a set called the domain) to exactly one output value (in a set called the codomain or range). On the other hand, a limit, describes the value that a function approaches as its input gets closer and closer to a specific value. Limit is just a type of function. The very definition of a limit is about the behavior of a function f(x) as its input x approaches a certain value a. Without a function, there's nothing to take the limit of. The notation $\lim_{x \to a}$ inherently implies that there is some function that depends on x that we are interested in observing. Therefore, the concept of a limit is connected to the existence of a function.

One-sided limits:

One-sided limits delve into the behavior of a function as its input approaches a specific value from only one direction: either from the left (smaller values) or from the right (larger values). We use specific notation to distinguish between these one-sided perspectives. The left-hand limit as x approaches a is denoted as $\lim_{x \to a^-}$ f(x) = $L^-$. The "-" signifies that x is approaching through a values strictly less than a. Similarly, the right-hand limit as x approaches a is denoted as $\lim_{x \to a^+}$ f(x) = $L^+$. The "+" signifies that x is approaching through a values strictly greater than a. The basic notation of Limit $\lim_{x \to a}$ f(x) = L (which is called two-sided limits) , is fundamentally linked to the behavior of its corresponding one-sided limits. There's a theorem says that the two-sided $\lim_{x \to a} f(x) \text{ exists if and only if } \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$. If the one-sided limits exist but are not equal ($L^- \neq L^+$), then the two-sided limit does not exist at x = a. One-sided limits become particularly important when dealing with functions that exhibit different behaviors on either side of a specific point. Let's look after an example. Suppose a function 

$f(x) = \begin{cases} 2x + 1 & \text{if } x < 3 \\ x^2 - 4 & \text{if } x \geq 3 \end{cases}$

And we want to find values of 'x' as it approaches to 3. If the value of 'x' is $3^-$, then the equation be $2x + 1$ because the value is less than 3. After doing the equation, we get 7 as the final answer. Again now if we take $3^+$ as 'x' value then the equation will be $x^2 - 4$. After doing the equation, we get 5 as the final answer. We can see the output of f(x) is different as the value approaches to $3^-$ and $3^+$. So the Limit does not exist but still one-sided limits do exist.

Suppose there exists a two-sided limit f(x) = $2x$ at both sides. Now suppose the want to examine how the function behaves as 'x' approaches to 3. Let's look this with a value table:

f(x) as 'x' approaches to $3^-$:

One-sided limit from left

x	f(x) = 2x
2.9	2(2.9) = 5.8
2.99	2(2.99) = 5.98
2.999	2(2.999) = 5.998

f(x) as 'x' approaches to $3^+$:

 

One-sided function from right

x	f(x) = 2x
3.1	2(3.1) = 6.2
3.01	2(3.01) = 6.02
3.001	2(3.001) = 6.002

From the value table, we can see that as x approaches 3 from both the left and the right, the values of f(x) = 2x approach 6. Therefore, the two-sided limit exists and is equal to 6:

$\lim_{x \to 3} 2x = 6$

Laws of Limits:

 Limit of a Constant: The limit of a constant function is the constant itself.

   $\lim_{x \to a} c = c$

   For example, $\lim_{x \to 5} 7 = 7$.

Limit of x: The limit of the identity function f(x) = x as x approaches a is a.

   $\lim_{x \to a} x = a$

   For example, $\lim_{x \to -2}$ $x = -2$.

Suppose two Limits $\lim_{x \to a}$$ f(x) = L$ and $\lim_{x \to a}$$ g(x) = M$ exist (where L and M are real numbers)

Limit of a Sum/Difference: The limit of a sum or difference of two functions is the sum or difference of their individual limits.

   $\lim_{x \to a} [f(x) \pm g(x)$] = $\lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$ = $L \pm M$

   For example, $\lim_{x \to 1} (x^2 + 3x)$ = $\lim_{x \to 1} x^2 + \lim_{x \to 1} 3x$ = $1^2 + 3(1)$ = 1 + 3 = 4.

Limit of a Product: The limit of a product of two functions is the product of their individual limits.

   $\lim_{x \to a} [f(x) \cdot g(x)]$ = $\lim_{x \to a} f(x) \cdot \lim_{x \to a}$$ g(x) = L \cdot M$

   For example, $\lim_{x \to 0} (x \cdot \cos(x))$ = $\lim_{x \to 0} x \cdot \lim_{x \to 0} \cos(x)$ = $0 \cdot 1 = 0.$

Limit of a Quotient: The limit of a quotient of two functions is the quotient of their individual limits, provided that the limit of the denominator is not zero.

   $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M}, \quad \text{if } M \neq 0$

   For example, $\lim_{x \to 2} \frac{x^2 + 1}{x + 3} = \frac{\lim_{x \to 2} (x^2 + 1)}{\lim_{x \to 2} (x + 3)} = \frac{2^2 + 1}{2 + 3} = \frac{5}{5} = 1$.

Limit of a Constant Multiple: The limit of a constant multiplied by a function is the constant multiplied by the limit of the function.

   $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x) = c \cdot L$

   For example, $\lim_{x \to -1} 4x^3 = 4 \cdot \lim_{x \to -1} x^3 = 4 \cdot (-1)^3 = 4 \cdot (-1) = -4.$

Limit of a Power: The limit of a function raised to a power n (where n is a positive integer) is the limit of the function raised to that power.

   $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n = L^n$

   For example, $\lim_{x \to 2} (x + 1)^2 = [\lim_{x \to 2} (x + 1)]^2 = (2 + 1)^2 = 3^2 = 9.$

I hope you liked the post. If you enjoyed it, please leave a comment which will inspire me to make more posts. Please note it's not end Limit is a big concept and I will cover them in another post.