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Find the values of k so that the function f is continuous at the indicated point in Exercises f x = k x ^ 2 if x < 2

26. Find the values of k so that the function f is continuous at the indicated point in Exercises

f (x) = \left\{\begin{matrix} kx^2 &if x \leq 2 \\ 3 & if x > 2 \end{matrix}\right. \: \: at \: \: x = 2

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Given function is
f (x) = \left\{\begin{matrix} kx^2 &if x \leq 2 \\ 3 & if x > 2 \end{matrix}\right.
When x = 2
For the function to be continuous
f(2) = R.H.L. = LH.L.
f(2) = 4k\\ \lim_{x\rightarrow 2^-}f(x)= 4k\\ \lim_{x\rightarrow 2^+}f(x) = 3\\ f(2) = \lim_{x\rightarrow 2^-}f(x) = \lim_{x\rightarrow 2^+}f(x)\\ 4k = 3\\ k = \frac{3}{4}
Hence,  the values of k so that the function f is continuous at x= 2 is \frac{3}{4}

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