JEE Main & Advanced Mathematics Functions Fundamental Theorems on Limits

The following theorems are very useful for evaluation of limits if \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=l\] and \[\underset{x\to 0}{\mathop{\lim }}\,g(x)=m\] (\[l\] and \[m\] are real numbers) then  

(1) \[\underset{x\to a}{\mathop{\lim }}\,(f(x)+g(x))=l+m\,\]                              (Sum rule)

(2) \[\underset{x\to a}{\mathop{\lim }}\,(f(x)-g(x))=l-m\]                  (Difference rule)    

(3) \[\underset{x\to a}{\mathop{\lim }}\,(f(x).g(x))=l.m\]                   (Product rule)                         

(4) \[\underset{x\to a}{\mathop{\lim }}\,k\,\,f(x)=k.l\]                                      (Constant multiple rule)        

(5) \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{f(x)}{g(x)}=\frac{l}{m},m\ne 0\]                             (Quotient rule)       

(6) If \[\underset{x\to a}{\mathop{\lim }}\,f(x)=+\infty \] or \[-\infty \], then \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{1}{f(x)}=0\]    

(7) \[\underset{x\to a}{\mathop{\lim }}\,\log \{f(x)\}=\log \,\{\underset{x\to a}{\mathop{\lim }}\,f(x)\}\]                   

(8) If \[f(x)\le g(x)\] for all \[x,\] then \[\underset{x\to a}{\mathop{\lim }}\,f(x)\le \underset{x\to a}{\mathop{\lim }}\,g(x)\]

(9) \[\underset{x\to a}{\mathop{\lim }}\,{{[f(x)]}^{g(x)}}={{\{\underset{x\to a}{\mathop{\lim }}\,f(x)\}}^{\underset{x\to a}{\mathop{\lim }}\,g(x)}}\]

(10) If \[p\] and \[q\] are integers, then \[\underset{x\to a}{\mathop{\lim }}\,{{(f(x))}^{p/q}}={{l}^{p/q}},\] provided \[{{(l)}^{p/q}}\] is a real number.

(11) If \[\underset{x\to a}{\mathop{\lim }}\,f(g(x))=f(\underset{x\to a}{\mathop{\lim }}\,g(x))=f(m)\] provided \['f'\] is continuous at \[g(x)=m.\,\,e.g.\]\[\underset{x\to a}{\mathop{\lim }}\,\ln [f(x)]=\ln (l),\]only if \[l>0.\]