Download App

Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability3 Marks
Question
$f(x)=\left\{\begin{array}{c}5 x^2-4, \text { if } x \leq 1 \\ 4 x^2-3 x \text {, if } x>2\end{array}\right.$ Examine the continuity.
✓

Answer

value of L.H.L. at $x=1$.$
\begin{aligned}
\lim _{x \rightarrow 1^{-}} f(x) & =\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0}[5(1-h)-4] \\
& =\lim _{h \rightarrow 0}(5-5 h-4)=\lim _{h \rightarrow 0}(1-5 h)=1
\end{aligned}
$
value of R.H.L. at $x=1$$
\begin{aligned}
\lim _{x \rightarrow 1^{+}} f(x) & =\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0}\left[4(1+h)^2-3(1+h)\right] \\
& =\lim _{h \rightarrow 0}\left(4+4 h^2+8 h-3-3 h\right) \\
& =\lim _{h \rightarrow 0}\left(4 h^2+5 h+1\right)=1
\end{aligned}
$value of function at $x=1$$
\begin{aligned}
& & f(x)=5 x-4 \\
\therefore & & f(1)=5 \times 1-4=5-4=1 \\
\because & & f(1)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0} f(1+h)
\end{aligned}
$
Hence function is continuous at $x=1$ and except $x=$ 1 this function is also continuous due to universal continuous.

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 "3 Marks Question" from Continuity and Differentiability→Continuity and Differentiability — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Compute $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big),$ if P(B) = 0.5 and $\text{P}(\text{A}\cap\text{B})=0.32$
→
$\int\frac{2\text{x}+3}{(\text{x}-1)^2}\text{dx}$
→
The following relation are defined on the set of real numbers.
aRb if $|\text{a}|\leq\text{b}$
Find whether these relation are reflexive, symmetric or transitive.
→
In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\text{k}+1,&\text{if}\text{ x}\leq5\\3\text{x}-5,&\text{if}\text{ x}>5\end{cases}\text{at x} =5$
→
Find the area of the triangle formed by O, A, B when $\overrightarrow{\text{OA}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\overrightarrow{\text{OB}}=-3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}.$
→
Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=2\sin\text{x}-\text{x}, -\frac{\pi}{2}\leq\text{x}\leq\frac{\pi}{2}$
→
Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$f(x) = x^3 - 3x$
→
Find the length of the perpendicular and coordinates of the foot of the perpendicular drawn from point $P(2,-1,5)$ on the given line $\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}$.
→
Find the direction cosines of the unit vector perpendicular to the plane $\vec{\text{r}}.\big(6\hat{\text{i}}-3\hat{\text{j}}-2\hat{\text{k}}\big)+1=0$ passing through the origin.
→
Show that the line through points $(4, 7, 8)$ and $(2, 3, 4)$ is parallel to the line throught the points $(-1, -2, 1)$ and $(1, 2, 5).$
→