Question

Find the relationship between 'a' and 'b' so that the function 'f' defined by 

\[f\left( x \right) = \begin{cases}ax + 1, & \text{ if }  x \leq 3 \\ bx + 3, & \text{ if } x > 3\end{cases}\] is continuous at x = 3.

 

Answer 1

Given: 

\[f\left( x \right) = \begin{cases}ax + 1, & \text{ if }  x \leq 3 \\ bx + 3, & \text{ if } x > 3\end{cases}\]

We have
(LHL at x = 3) = 

\[\lim_{x \to 3^-} f\left( x \right) = \lim_{h \to 0} f\left( 3 - h \right) = \lim_{h \to 0} a\left( 3 - h \right) + 1 = 3a + 1\]

(RHL at x = 3) = 

\[\lim_{x \to 3^+} f\left( x \right) = \lim_{h \to 0} f\left( 3 + h \right) = \lim_{h \to 0} b\left( 3 + h \right) + 3 = 3b + 3\]

\[If f\left( x \right)\text{  is continuous at x = 3, then } \]
\[ \lim_{x \to 3^-} f\left( x \right) = \lim_{x \to 3^+} f\left( x \right)\]
\[ \Rightarrow 3a + 1 = 3b + 3\]
\[ \Rightarrow 3a - 3b = 2\]

Hence, the required relationship between

\[a\text{and} is 3a - 3b = 2\]