KC Sinha Mathematics Solution Class 12 Chapter 9 संतता या सांतत्य (continuity) Exercise 9.1 (Q5-Q8)
Exercise 9.1
(Q1-Q4) (Q5-Q8) (Q9-Q12) (Q13-Q16) (Q17-Q20) (Q21-Q24) (Q25-Q28) (Q29-Q33)
Question 5
साचित करें कि f (x)=|x|, x=0 पर संतत है।
[Prove that
f (x)=|x| is continuous at x=0]
Sol :
$f(x)=\left\{\begin{array}{l}-x \text{ if x<0} \\ 0\text{ if x=0}\\x\text{ if x>0}\end{array}\right.$
At x=0
L.H.L
$\lim _{x \rightarrow 0^{-}} f(x)=\lim_{x\rightarrow0}(-x)$
=-0
=0
R.H.L
$\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0}(x)$
=0
f (0)=0
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)$
=
f (0)
f (x), x=0 पर संतता(continuous) है
Question 6
(क्या)Is $f(x)=(1+x)^{\frac{1}{x}}$ , when x≠0
f (x)=e , when x=0
x=0 पर संतत हैं ? (continuous at x=0 ?)
Sol :
At x=0
L.H.L
$\lim _{x \rightarrow 0^{-}} f(x)=\lim_{x\rightarrow0}(1+x)^{\frac{1}{x}}$
[if$\frac{1}{x} \rightarrow 0 \Rightarrow x \rightarrow \infty$]
$=\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$
R.H.L
$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}$
[if$\frac{1}{x} \rightarrow 0 \Rightarrow x \rightarrow \infty$]
$=\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$
f (0)=e
∴$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$
x=0 पर
f (x) संतता(continuous) है
Question 7
क्या
f (x) , x=0 पर संतत है? जहाँ
[If f (x) continuous at x=0 ? where]
$f(x)=\frac{\cos a x-\cos b x}{x^{2}}, \text{where }x\neq0 \left\{\begin{matrix}x<0\\x>0\end{matrix}\right.$
$f(x)=\frac{b^2-a^2}{2}, \text{where }x=0$
Sol :
At x=0 ,
L.H.L
=$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0} \frac{\cos a x-\cos b x}{x^{2}}$
[$\cos C-\cos D=2 \sin \frac{C+D}{2} \sin \frac{D-C}{2}$]
$=\lim _{x \rightarrow 0} \frac{2 \sin \frac{a x+b x}{2} \operatorname{sin} \frac{b x-a x}{2}}{x^{2}}$
$=2 \lim _{x \rightarrow 0} \frac{\sin \left(\frac{a+b}{2}\right) x}{x} \cdot \lim _{x \rightarrow 0} \frac{\sin \left(\frac{b-9}{2}\right) x}{x}$
$=2 \lim _{x \rightarrow 0} \frac{\operatorname{sin}\left(\frac{a+b}{2}\right) x}{\left(\frac{a+b}{2}\right)x} \times\left(\frac{a+b}{2}\right) \cdot \lim _{x \rightarrow 0} \frac{\sin \left(\frac{b-a}{2}\right)x}{\left(\frac{b-a}{2}\right) x}\times \left(\frac{b-a}{2}\right)$
$=2 \times\left(\frac{a+b}{x}\right) \times \frac{b-a}{2}$
$=\frac{b^{2}-a^{2}}{2}$
R.H.L
=$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} \frac{\cos a x-\cos bx}{x^{2}}$
$=\frac{b^{2}-a^{2}}{2}$
$f(0)=\frac{b^{2}-a^{2}}{2}$
∴$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$
f (x),x=0 संतता(continuous) है
Question 8
(यदि) If $f(x)=\left\{\begin{array}{ll}\frac{|x^3|}{x},\text{when }x\neq0;\left\{\begin{matrix}x<0\\x>0\end{matrix}\right. \\ 0,\text{when }x=0\end{array}\right.$
तो f(x) का x=0 पर सांतत्य की जाँच करें ।
[then test the continuity of
f (x) at x=0]
Sol :
At x=0 ,
L.H.L
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} \frac{\left|x^{3}\right|}{x}$
$=\lim _{x \rightarrow 0} \frac{-x^{2}}{x}=0^2$
=0
R.H.L
$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} \frac{| x^{-3}|}{x}$
$=\lim _{x \rightarrow 0} \frac{x^{3}}{x}=0^{2}$
=0
f (0)=0
∴ $\lim _{x \rightarrow 0^{-}} f(x)-\lim _{x \rightarrow 0^{+}} f(x)=f(0)$
f (x) ,x=0 संतता(continuous) है
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