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Find the values of k so that the function f is continuous at the indicated point in Exercises f x equals k x plus 1 , for x is lesser than 5 3x minus 5 for x greter than 5

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

 f (x) = \left\{\begin{matrix} kx +1 & if x \leq 5 \\ 3x-5 & if x > 5 \end{matrix}\right. \: \: at x = 5

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Given function is
f (x) = \left\{\begin{matrix} kx +1 & if x \leq 5 \\ 3x-5 & if x > 5 \end{matrix}\right.
When x = 5
For the function to be continuous
f(5) = R.H.L. = LH.L.
f(5) = 5k+1\\ \lim_{x\rightarrow 5^-}f(x)= 5k+1\\ \lim_{x\rightarrow 5^+}f(x) = 3(5)-5 = 15-5=10\\ f(5) = \lim_{x\rightarrow 5^-}f(x) = \lim_{x\rightarrow 5^+}f(x)\\ 5k+1 = 10\\ k = \frac{9}{5}
Hence,  the values of k so that the function f is continuous at x= 5 is \frac{9}{5}

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