Download App

Continuity — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity3 Marks
Question
A function f(x) is defined as, $​​​​​​​​​​\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-9}{\text{x}-3},&\text{if }\text{x}\neq3\\6,&\text{if }\text{ x}=3\end{cases}$ show that f(x) is continuous that x = 3
✓

Answer

Given,
$​​​​​​​​​​\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-9}{\text{x}-3},&\text{if } \text{ x}\neq3\\6 ,&\text{if }\text{ x}=3\end{cases}$
We observe 
$(\text{LHL at x }= 3)=\lim\limits_{\text{x} \rightarrow 3^-}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(3-\text{h})$
$=\lim\limits_{\text{h} \rightarrow 0}\frac{(3-\text{h})^2-9}{(3-\text{h})-3}=\lim\limits_{\text{h} \rightarrow 0}\frac{3^2-\text{h}^2-6\text{h}-9}{3-\text{h}-3}$
$=\lim\limits_{\text{h} \rightarrow 0}\frac{\text{h}^2+6\text{h}}{-\text{h}}=\lim\limits_{\text{h} \rightarrow 0}\frac{\text{h}(\text{h}-6)}{-\text{h}}$
$=\lim\limits_{\text{h} \rightarrow 0}(6-\text{h})=6$
$\text{(RHL at x = 3)}=\lim\limits_{\text{x} \rightarrow 3^+}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 3^+}\text{f}(3-\text{h})$
$=\lim\limits_{\text{h} \rightarrow 0}\frac{(3+\text{h})^2-9}{3+\text{h}-3}=\lim\limits_{\text{h} \rightarrow 0}\frac{3^2+\text{h}^2+6\text{h}-9}{\text{h}}$
$=\lim\limits_{\text{h} \rightarrow 0}\frac{\text{h}^2+6\text{h}}{\text{h}}=\lim\limits_{\text{h} \rightarrow 0}\frac{\text{h}(6+\text{h})}{\text{h}}=\lim\limits_{\text{h} \rightarrow 0}(6+\text{h})=6$
Given:
$\text{f}(3)=6$
$\therefore$ $\lim\limits_{\text{x} \rightarrow 3^-}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 3^+}\text{f}(\text{x})=\text{f}(3)$
Hence, f(x) is continuous at x = 3

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→Continuity — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

$\text{Let}\ \vec{\text{a}}=\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}},\ \vec{\text{b}}=3\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}$ and $ \vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$ Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{a}}\ \text{and}\ \vec{\text{b}},\ \text{and}\ \vec{\text{c}}\cdot\vec{\text{d}}=15.$
→
Integrate the function: $\frac{{2\cos x - 3\sin x}}{{6\cos x + 4\sin x}}$
→
Let $S$ be a relation on the set $R$ of all real numbers defined by $S = \{(a, b) \in R \times R: a^2 + b^2 = 1\}$ prove that $S$ is not an equivalence relation on $R.$
→
If $\text{y}=\frac{\log\text{x}}{\text{x}},$ show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{2\log\text{x}-3}{\text{x}^3}.$
→
Find gof and fog when $f : R \rightarrow R$ and $g : R \rightarrow R$ are defined by: $f(x) = 8x^3$ and $\text{g(x)}=\text{x}^\frac{1}{3}$
→
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
→
If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew symmetric.
→
Determine the value of $\lambda$ for which the following planes are perpendicular to other.
$2\text{x}-4\text{y}+3\text{z}=5$ and $\text{x}+2\text{y}+\lambda\text{z}=5$
→
A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.
→
Prove the following:
$\cos\Bigg(\sin^{-1}\frac{3}{5}+\cot^{-1}\frac{3}{2}\Bigg)=\frac{6}{5\sqrt{13}}$.
→