# JEE Main & Advanced Mathematics Functions Limit of a Function

## Limit of a Function

Category : JEE Main & Advanced

Let $y=f(x)$ be a function of $x$. If at $x=a,f(x)$ takes indeterminate form, then we consider the values of the function which are very near to $'a'$. If these values tend to a definite unique number as $x$ tends to $'a'$, then the unique number so obtained is called the limit of $f(x)$ at $x=a$ and we write it as $\underset{x\to a}{\mathop{\lim }}\,f(x)$.

(1) Left hand and right hand limit : Consider the values of the functions at the points which are very near to $a$ on the left of $a$. If these values tend to a definite unique number as $x$ tends to $a,$ then the unique number so obtained is called left-hand limit of $f(x)$ at $x=a$ and symbolically we write it as $f(a-0)=$$\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,\,f(x)=$$\,\underset{h\to 0}{\mathop{\lim }}\,\,f(a-h)$.

Similarly we can define right-hand limit of $f(x)$ at $x=a$ which is expressed as $f(a+0)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)$$=\underset{h\to 0}{\mathop{\lim }}\,f(a+h)$.

(2) Method for finding L.H.L. and R.H.L.

(i) For finding right hand limit (R.H.L.) of the function, we write $x+h$ in place of $x,$ while for left hand limit (L.H.L.) we write $x-h$ in place of $x$.

(ii) Then we replace $x$ by $'a'$ in the function so obtained.

(iii) Lastly we find limit $h\to 0$.

(3) Existence of limit : $\underset{x\to a}{\mathop{\lim }}\,f(x)\,\,$exists when,

(i) $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)$ and $\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)$ exist i.e. L.H.L. and R.H.L. both exists.

(ii) $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)$ i.e. L.H.L. = R.H.L.

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