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LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES4 Marks
Question
Suppose $\text{f}(\text{x})=\begin{cases}\text{a+bx},& \text{x}< 0\\4 & \text{x} = 1\\\text{b-ax},&\text{x}>1\end{cases}$and if $\lim\limits{\text{x}\rightarrow1}=\text{f}(1)$ what are possible values of a and b?
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Answer

The given function is $\text{f}(\text{x})=\begin{cases}\text{a+bx},& \text{x}< 0\\4 & \text{x} = 1\\\text{b-ax},&\text{x}>1\end{cases}$ $=\lim\limits_{\text{x}\rightarrow1^-}\text{f}(\text{x})=\lim\limits_{\text{x}\rightarrow1}(\text{a+bx})=\text{a+b}$ $=\lim\limits_{\text{x}\rightarrow1^+}\text{f}(\text{x})==\lim\limits_{\text{x}\rightarrow1}(\text{b-ax})=\text{b-a}$ $=\text{f}(1)=4$It is given that $\lim\limits_{\text{1}\rightarrow1}\text{f}(\text{x})=\text{f}(1)$
$\therefore=\lim\limits_{\text{x}\rightarrow1^-}\text{f}(\text{x})=\lim\limits_{\text{x}\rightarrow1^+}\text{f}(\text{x})=\lim\limits_{\text{x}\rightarrow1}\text{f}(\text{x})=\text{f}(1)$
$\Rightarrow\text{a+b}=4$and$\text{b-a}=4$ On solving these two equations, we obtain a=0 nd b=4. Thus, the respective possible values of a and b are 0 and 4.

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