Let f(a)=g(a)=k and their nth derivatives f(n)(a),g(n)(a) exist and are not equal for some n. If
\[\lim_{x\to a}\frac{f(a)g(x)-f(a)-g(a)f(x)+f(a)}{g(x)-f(x)}=4\]
then the value of k is
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Official solution
\[\mathop {\lim }\limits_{x \to a} {{f\left( a \right)g'\left( x \right) - g\left( a \right)f'\left( x \right)} \over {g'\left( x \right) - f'\left( x \right)}}\]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\] (By L′ Hospital rule)
\[\mathop {\lim }\limits_{x \to a} {{k\,\,g'\left( x \right) - k\,\,f'\left( x \right)} \over {g'\left( x \right) - f'\left( x \right)}} = 4\]
\[\therefore\] k=4.