Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{f}\text{(x)}=\begin{cases}\text{e}^\frac{1}{\text {x}}, & \text{if} \text{ x}\neq 0\\1, & \text{if}\text{x} = 0\end{cases}$ find whethe f is continuous at x = 0.
✓

Answer

Given,
$\text{f}\text{(x)}=\begin{cases}\text{e}^\frac{1}{\text {x}}, & \text{if} \text{ x}\neq 0\\1, & \text{if}\text{x} = 0\end{cases}$
We observe
$(\text{LHL at x}=0)=\lim\limits_{\text{x} \rightarrow 0^-}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(0-\text{x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}\text{(-h)}$
$=\lim\limits_{\text{h} \rightarrow 0}\text{e}^\frac{-1}{\text{h}}=\lim\limits_{\text{h} \rightarrow 0}\bigg(\frac{1}{\text{e}^{\frac{1}{\text{h}}}}\bigg)=\frac{1}{\lim\limits_{\text{h} \rightarrow 0}\text{e}^{\frac{1}{\text{h}}}}=0$
$(\text{RHL at x}=0)=\lim\limits_{\text{h} \rightarrow 0^+}\text{f}(\text{x})=\lim\limits_{\text{h} \rightarrow 0}\text{f}(\text{h})$
$=\lim\limits_{\text{h} \rightarrow 0^+}\text{e}^\frac{1}{\text{h}}=\infty$
Given,
$\text{f}(0)=1$
It is known for a function f(x) to be continuous at x = a,
$\lim\limits_{\text{x} \rightarrow \text{a}^-}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow \text{a}^+}\text{f}\text{(x)}=\text{f}\text{(a)}$
But here,
$\lim\limits_{\text{x} \rightarrow \text{a}^-}\text{f}\text{(x)}\neq\lim\limits_{\text{x} \rightarrow \text{0}^+}\text{f}\text{(x)}$
Hence, f(x) is discontinuous at x = 0.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^1_{-1}|\text{x}\cos\pi\text{x}|\text{dx}$
→
Evalute the following integrals:
$\int\frac{\cos\text{x}}{2+3\sin\text{x}}\text{dx}$
→
Show that $\text{f}(\text{x})=\tan^{-1}(\sin\text{x}+\cos\text{x})$ is a decreasing function on the interval $\Big(\frac{\pi}{4},\frac{\pi}{2}\Big).$
→
Find the equation of the plane through (2, 3, -4) and (1, -1, 3) and parallel to x-axis.
→
Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.
→
If $(a x+b) e^{y / x}=x$ then show that : $ x^3 \frac{d^2 y}{d x^2}=\left(x \frac{d y}{d x}-y\right)^2$
→
Find the value of a and b so that the function f(x) defind by $\text{f(x)}=\begin{cases}\text{x}+\text{a}\sqrt{2}\sin\text{x},&\text{if }0\leq\text{x}<\frac{\pi}{4}\\2\text{x}\cot\text{ x}+\text{b},&\text{if }\frac{\pi}{4}\leq\text{x}<\frac{\pi}{2}\\\text{a}\cos2\text{x}-\text{b}\sin\text{x},&\text{if }\frac{\pi}{2}\leq\text{x}\leq\pi\end{cases}$ becomes continuous on $[0,\pi]$
→
Find the value of k if the function: $f (x) = \begin{matrix} kx^{2}& x\geq 1 & \\ 4 & x< 1 & \\ \end{matrix} \text{is continuous at x = 1} $
→
A company manufactures two types of screws A and B. All the screws have to pass through a threading machine and a slotting machine. A box of Type A screws requires 2 minutes on the threading machine and 3 minutes on the slotting machine. A box of type B screws requires 8 minutes of threading on the threading machine and 2 minutes on the slotting machine. In a week, each machine is available for 60 hours.
On selling these screws, the company gets a profit of Rs 100 per box on type A screws and Rs 170 per box on type B screws.
→
$\int\sin\text{x}\sqrt{1-\cos2\text{x}}\text{ dx}$
→