Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find which of the function:
$\text{f(x)}=\begin{cases}\frac{2\text{x}^2-3\text{x}-2}{\text{x}-2},&\text{if x}\neq2\\5,&\text{if x}=2\end{cases}$
at x = 2
✓

Answer

We have, $\text{f(x)}=\begin{cases}\frac{2\text{x}^2-3\text{x}-2}{\text{x}-2},&\text{if x}\neq2\\5,&\text{if x}=2\end{cases}$ at x = 2.
At x = 2, $\text{L.H.L}=\lim\limits_{\text{x}\rightarrow2^-}\frac{2\text{x}^2-3\text{x}-2}{\text{x}-2}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{2(2-\text{h})^2-3(2-\text{h})-2}{(2-\text{h})-2}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{8+2\text{h}^2-8\text{h}-6+3\text{h}-2}{-\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{2\text{h}^2-5\text{h}}{-\text{h}}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}(2\text{h}-5)}{-\text{h}}=5$
$\text{R.H.L}=\lim\limits_{\text{h}\rightarrow2^+}\frac{2\text{x}^2-3\text{x}-2}{\text{x}-2}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{2(2+\text{h})^2-3(2+\text{h})-2}{(2+\text{h})-2}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{8+2\text{h}^2+8\text{h}-6-3\text{h}-2}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{2\text{h}^2+5\text{h}}{\text{h}}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}(2\text{h}+5)}{\text{h}}=5$
and f(2) = 5
$\therefore$ L.H.L = R.H.L = f(2)
So, f(x) is continuous at x = 2.

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

If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), show that at $\text{t}=\frac{\pi}{4},\big(\frac{\text{dy}}{\text{dx}}\big)=\frac{\text{b}}{\text{a}}$
→
If $\text{y}=\sin\Big[2\tan^{-1}\Big\{\sqrt{\frac{1-\text{x}}{1+\text{x}}}\Big\}\Big],$ find $\frac{\text{dy}}{\text{dx}}.$
→
In each of the show that the given differential equation is homogeneous and solve each of them.
$\text{x dy}-\text{y dx}=\sqrt{\text{x}^2+\text{y}^2}\ \text{dx}$
→
If $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+5\hat{\text{j}},\ 3\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}-6\hat{\text{j}}-\hat{\text{k}}$ respectively are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether $\vec{\text{AB}}$ and $\vec{\text{CD}}$ are collinear or not.
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = \sec(\text{x}+\text{y})$
→
Evaluate the following intregals:$\int\frac{1}{3+2\cos^2\text{x}}\text{ dx}$
→
Solve the following systems of linear equations by cramer's rule:
9x + 5y = 10,
3x - 2y = 8
→
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is $\frac{4\text{r}}{3}.$ Also find the maximum volume of cone.
→
Using properties of determinants, prove that:
$\begin{vmatrix}3\text{a}&-\text{a+b}&-\text{a+c}\\-\text{b+a}&3\text{b}&-\text{b+c}\\-\text{c+a}&\text{c+b}&3\text{c}\end{vmatrix}=3 (\text{a + b + c}) (\text{ab + bc + ca})$
→
Find the equation of a plane which is at a distance of $3\sqrt{3}\text{ units}$ from the origin and the normal to which is equally inclined to the coordinate axes.
→