A) \[2\]
B) \[4\]
C) \[-2\]
D) \[-4\]
Correct Answer: B
Solution :
Given that \[\underset{x\to 9}{\mathop{\lim }}\,\frac{f(x)-9}{x-9}=4\] \[\Rightarrow \] \[\underset{x\to 9}{\mathop{\lim }}\,\frac{{{(\sqrt{f(x)})}^{2}}-{{(3)}^{2}}}{{{(\sqrt{x})}^{2}}-{{3}^{2}}}=4\] \[\Rightarrow \] \[\lim \frac{(\sqrt{f(x)}-3)}{(\sqrt{x}-3)}\times \underset{x\to 9}{\mathop{\lim }}\,\frac{\sqrt{f(x)}+3}{\sqrt{x}+3}=4\] \[\Rightarrow \] \[\underset{x\to 9}{\mathop{\lim }}\,\frac{(\sqrt{f(x)}-3)}{(\sqrt{x}-3)}\times \frac{\sqrt{f(9)}+3}{3+3}=4\] \[\Rightarrow \] \[\lim \frac{(\sqrt{f(x)}-3)}{(\sqrt{x}-3)}\times \frac{6}{6}=4\] \[[\because \,\,f(9)=9\]given] \[\Rightarrow \] \[\underset{x\to 9}{\mathop{\lim }}\,\frac{(\sqrt{f(x)}-3)}{(\sqrt{x}-3)}=4\]
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