JEE Main & Advanced Mathematics Functions Limit of a Function

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.