In mathematical terms, the limit of a function \( f(x) \) as \( x \) approaches a point \( c \) is given by
\[
\lim_{{x \to c}} f(x) = L
\]
Here, \( L \) is the value that \( f(x) \) approaches as \( x \) gets closer to \( c \).
The limit of f (x) as x tends to x is to be thought of as the value f (x) should assume at x = c
Another Definition
The formal definition of a limit is as follows: Let \( f(x) \) be a function defined on some open interval containing \( c \), except possibly at \( c \) itself. We say that the limit of \( f(x) \) as \( x \) approaches \( c \) is \( L \), written as
\[
\lim_{{x \to c}} f(x) = L
\]
if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \epsilon \).
Left-hand Limit
The left-hand limit of \( f(x) \) as \( x \) approaches \( c \) is denoted by
\[
\lim_{{x \to c^-}} f(x)
\]
It is the expected value of f at x = a given the values of f near x to the left of a.
Right-hand Limit
The right-hand limit of \( f(x) \) as \( x \) approaches \( c \) is denoted by
\[
\lim_{{x \to c^+}} f(x)
\]
Existence of Limit
For a limit to exist, the left-hand and right-hand limits must be equal, i.e.,
\[
\lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x) = \lim_{{x \to c}} f(x)
\]
Algebra of Limits
1. Limit of a Sum: Limit of sum of two functions is sum of the limits of the functions
\( \lim_{{x \to c}} [ f(x) + g(x) ] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x) \)
2.Limit of Difference: Limit of difference of two functions is sum of the limits of the functions
\( \lim_{{x \to c}} [ f(x) - g(x) ] = \lim_{{x \to c}} f(x) - \lim_{{x \to c}} g(x) \)
3. Limit of a Product : Limit of product of two functions is product of the limits of the functions
\( \lim_{{x \to c}} [ f(x) \times g(x) ] = \lim_{{x \to c}} f(x) \times \lim_{{x \to c}} g(x) \)
4 if g(x)=k,then
\( \lim_{{x \to c}} [ f(x) \times k ] = k \lim_{{x \to c}} f(x) \)
5. Limit of a Quotient: Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero)
\( \lim_{{x \to c}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to c}} f(x)}{\lim_{{x \to c}} g(x)} \) (provided \( \lim_{{x \to c}} g(x) \neq 0 \))
Types of Limits
Finite Limits
These are limits where the function approaches a particular real number as \( x \) approaches a specific value. For example,
\[
\lim_{{x \to 2}} (x^2 - 4) = 0
\]
Infinite Limits
Here, the function approaches infinity as \( x \) approaches a specific value. For example,
\[
\lim_{{x \to 0}} \frac{1}{x^2} = \infty
\]
Limits at Infinity
These limits describe the behavior of a function as \( x \) goes to infinity. For example,
\[
\lim_{{x \to \infty}} \frac{1}{x} = 0
\]
Limit of Polynomial Function
We know that
\[
\lim_{{x \to a}} x = a
\]
Now
\[
\lim_{{x \to a}} x^2 = \lim_{{x \to a}} x.x = \lim_{{x \to a}} x \lim_{{x \to a}} x= a.a=a^2
\]
Similarly
\[
\lim_{{x \to a}} x^n = a^n
\]
A polynomial function f(x) in one variable x is an algebraic expression in x term as
$f(x)=a_n x^n+a_(n-1) x^(n-1)+a_(n-2) x^(n-2)+⋯………+ax+a_0$
Where $a_n,a_{n-1},....,a,a_0$ are constant and real numbers and $a_n$ is not equal to zero
Limit is calculated as
\[
\lim_{{x \to a}} f(x) =\lim_{{x \to a}} a_n x^n+a_(n-1) x^(n-1)+a_(n-2) x^(n-2)+⋯………+ax+a_0
\]
\[
=a_n \lim_{{x \to a}} x^n + a_(n-1) \lim_{{x \to a}} x^(n-1)+ a_(n-2) \lim_{{x \to a}} x^(n-2)+⋯………+ a \lim_{{x \to a}} x+a_0 \lim_{{x \to a}}
\]
\[
=a_n a^n + a_(n-1) a^(n-1)+ a_(n-2) a^(n-2)+⋯………+ a_1 a+a_0 =f(a)
\]
Hence
\[
\lim_{{x \to a}} f(x) = f(a)
\]
Limit of Rational Function
Rational Function is defined as
$f(x)=\frac {g(x)}{h(x)}$
Where g(x) and h(x) are polynomial function and $h(x) \ne 0$
\[
\lim_{{x \to a}} \frac {g(x)}{h(x)} = \frac { \lim_{{x \to a}} g(x)}{\lim_{{x \to a}} h(x)}= \frac {g(a)}{h(a)}
\]
Case A if h(a) =0 and g(a) =0
Now we can write
$g(x) = (x-a)^k g' (x) $
$h(x) = (x-a)^l h' (x) $
if k > l
If k< l
Limit is undefined Case B - h(a) =0 and $g(a) \ne 0$
Then Limit is undefined
A general rule that needs to be kept in mind while evaluating limits is the following.
(i) First we check the value of f (a) and g(a).
(ii)If both are 0, then we see if we can get the factor which is causing the terms to vanish, i.e., see if we can write $f(x) = f_1(x) f_2(x)$ so that f1(a) = 0 and $f_2(a) \ne 0$.
(iii)Similarly, we write $g(x) = g_1(x) g_2(x)$, where $g_1(a) = 0$ and $g_2(a) \ne 0$.
(iv) Cancel out the common factors from f(x) and g(x) (if possible) and write
$\frac {f(x)}{g(x)} = \frac {p(x)}{q(x)}$
Theorem II
Let f and g be two real valued functions with the same domain such that $f (x) \leq g( x)$ for all x in the domain of definition,
For some a, if both
\[
\lim_{{x \to a}} f(x)
\]
and
\[
\lim_{{x \to a}} g(x)
\]
exists, then
$\lim_{{x \to a}} f(x) \leq \lim_{{x \to a}} g(x)$
Sandwitch Theorem
Let f, g and h be real functions such that
$f (x) \leq g( x) \leq h(x)$ for all x in the common domain of definition.
For some real number a, if
$\lim_{{x \to a}} f(x) =l=\lim_{{x \to a}} h(x)$
then
$\lim_{{x \to a}} g(x) =l$
Limits of Trigonometric Function
we can easily the below relation geometrically
$cos x < \frac {sin x}{x} < 1$ for $ 0 < |x| < \frac {\pi}{2}$
Lets define the limit of two important Theorem based on Above Formula and Sandwitch theorem
Theorem I
$\lim_{{x \to 0}} \frac {sin x}{x} =1$ Proof
$\lim_{{x \to 0}} cos x =1$
$\lim_{{x \to 0}} 1 =1$
Therefore from Sandwitch theorem
$\lim_{{x \to 0}} \frac {sin x}{x} =1$
(a)A limit is undefined if the function doesn't approach a specific value or if it approaches different values from the left and right.
(b) Limits can be calculated using a variety of methods including direct substitution, factorization, rationalization,Theorems stated abive and using L'Hôpital's Rule for indeterminate forms.
