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  • If the function <img alt="\mathrm{f(x)=\left\{\begin{array}{ccc} \frac{3}{x^2} \sin 2 x^2 & ; & \text { if } x<0 \\ 0 & ; & \text { if } x=2 \\ \frac{x^2-3 x+k}{1-3 x^2} &am

If the function \mathrm{f(x)=\left\{\begin{array}{ccc} \frac{3}{x^2} \sin 2 x^2 & ; & \text { if } x<0 \\ 0 & ; & \text { if } x=2 \\ \frac{x^2-3 x+k}{1-3 x^2} & ; & \text { if } x \geq 0, x \neq 2 \end{array}\right.} is continuous at x = 0, then the value of k is 

 

Option: 1

4


Option: 2

5


Option: 3

6


Option: 4

none of these


Answers (1)

best_answer

\mathrm{\begin{aligned} \lim _{x \rightarrow 0-} f(x) & =\lim _{x \rightarrow 0} \frac{3}{x^2} \sin \left(2 x^2\right) \\ & =\lim _{x \rightarrow 0} \frac{6}{2 x^2} \sin \left(2 x^2\right)=6 \end{aligned}}

            \mathrm{\lim _{x \rightarrow 0+} f(x)=\lim _{x \rightarrow 0} \frac{x^2-3 x+k}{1-3 x^2}=k}

Since \mathrm{f(x)}  is continuous at x = 0

\mathrm{\begin{array}{ll} \therefore & \lim _{x \rightarrow 0-} f(x)=\lim _{x \rightarrow 0+} f(x)=f(0) \\ \therefore & k=6 \end{array}}

 

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manish

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