Question

$\lim_{x\rightarrow0}f(x) \,and \,\lim_{x\rightarrow1}f(x), \,ज्ञात कीजिए , जहाँ \,f(x) = \right \begin{cases}{2x + 3}, \,\,\,\,\,x \leq 0\atop 3(x + 1), \, x \ \textgreater \ 0 \end{cases}$

Answer 1

Answer:

Step-by-step explanation:

दी गई सीमा   x → 0

तथा  $f(x) = \right \begin{cases}{2x + 3}, \,\,\,\,\,x \leq 0\atop 3(x + 1), \, x \  \textgreater \  0 \end{cases}$

x > 0  के लिए -

$R.H.L.=\lim_x_\rightarrow_0^+f(x)=\lim_x_\rightarrow_0^+3(x+1)\\\\=3(0+1)=3$

x <  0  के लिए  -

$L.H.L.=\lim_x_\rightarrow_0^-f(x)=\lim_x_\rightarrow_0^-(2x+3)\\\\=(2*(0)+3)=3$

$\lim_x_\rightarrow_0^+f(x)=\lim_x_\rightarrow_0^-f(x)$

∴$\lim_x_\rightarrow_0f(x)=3$

अब सीमा   x → 1

x > 1  के लिए -

$R.H.L.=\lim_x_\rightarrow_1^+f(x)=\lim_x_\rightarrow_1^+3(x+1)\\\\=3(1+1)=6$

x < 1  के लिए  -

$L.H.L.=\lim_x_\rightarrow_1^-f(x)=\lim_x_\rightarrow_1^-(2x+3)\\\\=(2*(-1)+3)=1$

∵ $\lim_x_\rightarrow_1^+f(x)\neq \lim_x_\rightarrow_1^+f(x)$

$\lim_x_\rightarrow_1f(x)$का अस्तित्व नहीं है।