CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
Show that f(x) = |x - 5| is continuous but not differentiable at x = 5.
✓
Answer
A function f is a differentiable function if and only if $\text{Lf (c)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f(a}-\text{h})-\text{f}(\text{a})}{-\text{h}}$ and $\text{Rf}'(\text{c})=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{a}-\text{h})-\text{f(a)}}{\text{h}}$ are equal. Consider, f(x) = |x - 5| $\therefore\ \text{f(x)}=\begin{cases}-(\text{x}-5),&\text{if x}<5\\\text{x}-5,&\text{if x}\geq5\end{cases}$ For continuity at x = 5, $\text{L.H.L}=\lim\limits_{\text{x}\rightarrow5^-}(-\text{x}+5)$ $=\lim\limits_{\text{h}\rightarrow0}\big[-(5-\text{h})+5\big]=\lim\limits_{\text{h}\rightarrow0}\text{h}=0$ $\text{R.H.L}=\lim\limits_{\text{h}\rightarrow5^+}(\text{x}-5)$ $=\lim\limits_{\text{h}\rightarrow0}(5+\text{h}-5)=\lim\limits_{\text{h}\rightarrow0}\text{h}=0$ $\text{f}(5)=5-5=0$ $\Rightarrow\ \text{L.H.L}=\text{R.H.L}=\text{f}(5)$ Hence, f(x) is continuous at x = 5 Now, $\text{Lf}'(5)=\lim\limits_{\text{x}\rightarrow5^-}\frac{\text{f(x)}-\text{f}(5)}{\text{x}-5}$ $=\lim\limits_{\text{x}\rightarrow5^-}\frac{-\text{x}+5-0}{\text{x}-5}=-1$ $\text{Rf}'(5)=\lim\limits_{\text{x}\rightarrow5^+}\frac{\text{f(x)}-\text{f}(5)}{\text{x}-5}$ $=\lim\limits_{\text{x}\rightarrow5^+}\frac{\text{x}-5-0}{\text{x}-5}=1$ Lf'(5) ≠ Rf'(5) So, f(x) = |x - 5| is not differentiable at x = 5.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.