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Sum of n Terms

Find the valu...

Question

Find the values of $k$ so that the function $f$ is continuous at the indicated point:
$f (x) =$ ${\begin{matrix} k x + 1, i f x \leq 5 3 x - 5, i f x > 5 \end{matrix}$ at $x = 5$

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Solution

The given function is $f (x) = {\begin{matrix} k x + 1, if x \leq 5 3 x - 5, if x > 5 \end{matrix}$
The given function $f$ is continuous at $x = 5$ , if $f$ is defined at $x = 5$ and if the value of $f$ at $x = 5$ equals the limit of $f$ at $x = 5$
It is evident that $f$ is defined at $x = 5$ and $f (5) = k x + 1 = 5 k + 1$
$lim x \to 5^{-} f (x) = lim x \to 5^{+} f (x) = f (5)$
$\Rightarrow lim x \to 5 (k x + 1) = lim x \to 5 (3 x - 5) = 5 k + 1$
$\Rightarrow 5 k + 1 = 15 - 5 = 5 k + 1$
$\Rightarrow 5 k + 1 = 10$
$\Rightarrow 5 k = 9 \Rightarrow k = \frac{9}{5}$
Therefore, the required value of $k$ is $\frac{9}{5}$

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