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  • Find the value of k so that the function f is continuous at the indicated point: f(x) = 3x-8

Find the value of k so that the function f is continuous at the indicated point:

f(x)=\begin{array}{cl} 3 x-8, & \text { if } x \leq 5 \\ 2 k, & \text { if } x>5 \end{array} \text { at } x=5

Answers (1)

Given,

f(x)=\begin{array}{cl} 3 x-8, & \text { if } x \leq 5 \\ 2 k, & \text { if } x>5 \end{array} \text { at } x=5

We need to find the value of k such that f(x) is continuous at x = 5.
A function f(x) is said to be continuous at x = c if,

Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)

Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 5.

\lim _{h \rightarrow 0} f(5-h)=\lim _{h \rightarrow 0} f(5+h)=f(5)

As we have to find k so pick out a combination so that we get k in our equation.
In this question we take LHL = f(5)

\\ \therefore \lim _{h \rightarrow 0} f(5-h)=f(5) \\ \Rightarrow \lim _{h \rightarrow 0}\{3(5-h)-8\}=2 k \\ \Rightarrow 3(5-0)-8=2 k \\ \Rightarrow 15-8=2 k \\ \Rightarrow 2 k=7 \\ \therefore k=7 / 2

Posted by

infoexpert22

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