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Continuity — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity5 Marks
Question
Find the relationship between 'a' and 'b' so that the function 'f' defind by $\text{f(x)}=\begin{cases}\text{ax}+1,&\text{if }\text{ x}\leq3\\\text{bx}+3,&\text{if }\text{ x}>3\end{cases}$ is continuous at x = 3.
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Answer

Given,
$\text{f(x)}=\begin{cases}\text{ax}+1,&\text{if }\text{ x}\leq3\\\text{bx}+3,&\text{if }\text{ x}>3\end{cases}$
We have,
$(\text{LHL at x}= 3)=\lim_\limits{\text{x}\rightarrow3^-}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}\text{f}(3-\text{h})$
$=\lim_\limits{\text{h}\rightarrow0}\text{a}(3-\text{h})+1=3\text{a}+1$
$(\text{RHL at x}= 3)=\lim_\limits{\text{x}\rightarrow3^+}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}\text{f}(3+\text{h})$
$=\lim_\limits{\text{h}\rightarrow0}\text{b}(3+\text{h})+3=3\text{b}+3$
If f(x) is continuous at x = 3, then
$\lim_\limits{\text{x}\rightarrow3^-}\text{f(x)}=\lim_\limits{\text{x}\rightarrow3^+}\text{f(x)}$
$\Rightarrow3\text{a}+1=3\text{b}+3$
$\Rightarrow3\text{a}-3\text{b}=2$
Hence, the required relationship between a & b is 3a - 2b = 2

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