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.

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

Find the Relationship Between 'A' and 'B' So that the Function 'F' Defined by F ( X ) = { a X + 1 , If X ≤ 3 B X + 3 , If X > 3 is Continuous at X = 3. - Mathematics

Question

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

f (x) = {\begin{cases} a x + 1, & if x \leq 3 \\ b x + 3, & if x > 3 \end{cases}

is continuous at x = 3.

Sum

Solution

Given:

f (x) = {\begin{cases} a x + 1, & if x \leq 3 \\ b x + 3, & if x > 3 \end{cases}

We have
(LHL at x = 3) =

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

(RHL at x = 3) =

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

$I f f (x) is continuous at x = 3, then$
$lim_{x \to 3^{-}} f (x) = lim_{x \to 3^{+}} f (x)$
$\Rightarrow 3 a + 1 = 3 b + 3$
$\Rightarrow 3 a - 3 b = 2$

Hence, the required relationship between

a and i s 3 a - 3 b = 2

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Concept of Continuity

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Q 46 Q 45 Q 1

Chapter 9: Continuity - Exercise 9.1 [Page 21]

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Chapter 9 Continuity
Exercise 9.1 | Q 46 | Page 21

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