x tends to infinity সংক্রান্ত
The value of limx→∞x(x{ln(x)−ln(x+1)}+1)\displaystyle\lim_{x\rightarrow \infty}x(x\{ln (x)-ln (x+1)\}+1)x→∞limx(x{ln(x)−ln(x+1)}+1) is?
1
1e\dfrac{1}{e}e1
12\dfrac{1}{2}21
14\dfrac{1}{4}41
limx→∞x[x{lnx−ln(x+1)}=limx→∞x[x{ln(xx+1)+1}]=limx→∞x[x{lnxx(1+1x)}+1]=limx→∞x[x{ln(11+1x)+1]=limx→∞x[x{ln(1+1x)−1}+1] \begin{array}{l}\lim _{x \rightarrow \infty} x[x\{\ln x-\ln (x+1)\} \\ =\lim _{x \rightarrow \infty}x\left[x\left\{\ln \left(\frac{x}{x+1}\right)+1\right\}\right] \\ =\lim _{x \rightarrow \infty} x\left[x\left\{\ln \frac{x}{x\left(1+\frac{1}{x}\right)}\right\}+1\right] \\ =\lim _{x \rightarrow \infty} x\left[x\left\{\ln \left(\frac{1}{1+\frac{1}{x}}\right)+1\right]\right. \\ =\lim _{x \rightarrow \infty} x\left[x\left\{\ln \left(1+\frac{1}{x}\right)^{-1}\right\}+1\right] \\\end{array} limx→∞x[x{lnx−ln(x+1)}=limx→∞x[x{ln(x+1x)+1}]=limx→∞x[x{lnx(1+x1)x}+1]=limx→∞x[x{ln(1+x11)+1]=limx→∞x[x{ln(1+x1)−1}+1]
=limx→∞x[xln(1+1x)−1+1]=limx→∞x[1+ln(1+1x)−x]=limx→∞x[lne−ln(1+1x)x]=limx→∞[xlne−xlne]=0 \begin{array}{l}=\lim _{x \rightarrow \infty}x\left[x \ln \left(1+\frac{1}{x}\right)^{-1}+1\right] \\ =\lim _{x \rightarrow \infty} x\left[1+\ln \left(1+\frac{1}{x}\right)^{-x}\right] \\ =\lim _{x \rightarrow \infty} x\left[\ln e-\ln \left(1+\frac{1}{x}\right)^{x}\right] \\ =\lim _{x \rightarrow \infty}[x \operatorname{ln} e-x \operatorname{ln } e]=0\end{array} =limx→∞x[xln(1+x1)−1+1]=limx→∞x[1+ln(1+x1)−x]=limx→∞x[lne−ln(1+x1)x]=limx→∞[xlne−xlne]=0
limx→∞(1+1x)2=1 \begin{array}{l}\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{2} \\ =1\end{array} limx→∞(1+x1)2=1
Ai এর মাধ্যমে
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প্র্যাকটিস এর মাধ্যমে নিজেকে তৈরি করে ফেলো
উত্তর দিবে তোমার বই থেকে ও তোমার মত করে।
সারা দেশের শিক্ষার্থীদের মধ্যে নিজের অবস্থান যাচাই
If f(x)=13(f(x+1)+5f(x+2))f(x) = \dfrac {1}{3}\left (f (x + 1) + \dfrac {5}{f(x + 2)}\right )f(x)=31(f(x+1)+f(x+2)5) and f(x)>0f(x) > 0f(x)>0 for all xϵRx \epsilon RxϵR, then limx→∞f(x)\displaystyle \lim_{x\rightarrow \infty} f(x)x→∞limf(x) is
f(x)=2xx+1 হলে- f(x)=\frac{2 x}{x+1} \text { হলে- } f(x)=x+12x হলে-
i. limx→∞f(x)=2 \lim _{x \rightarrow \infty} f(x)=2 limx→∞f(x)=2
ii. ddx[f(x)]=2(x+1)2 \frac{\mathrm{d}}{\mathrm{dx}}[f(\mathrm{x})]=\frac{2}{(\mathrm{x}+1)^{2}} dxd[f(x)]=(x+1)22
iii. limx→2f(x)=f(2) \lim _{\mathrm{x} \rightarrow 2} f(\mathrm{x})=f(2) limx→2f(x)=f(2)
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