লিমিট
নিচের সীমার মান কোনটি? limx→ysinx−sinyx−y\lim _{x \rightarrow y} \frac{\sin x-\sin y}{x-y}limx→yx−ysinx−siny
cot x
cosec x
cos y
sin y
limx→ysinx−sinyx−y=limx→y2sinx−y2cosx+y2x−y=2.limx→ysinx−y2x−y2×12×limx→ycosx+y2=2×1×12cosy+y2=cosy (Ans.) \begin{array}{l} \lim _{x \rightarrow y} \frac{\sin x-\sin y}{x-y} \\ =\lim _{x \rightarrow y} \frac{2 \sin \frac{x-y}{2} \cos \frac{x+y}{2}}{x-y} \\ =2 . \lim _{x \rightarrow y} \frac{\sin \frac{x-y}{2}}{\frac{x-y}{2}} \times \frac{1}{2} \times \lim _{x \rightarrow y} \cos \frac{x+y}{2} \\ =2 \times 1 \times \frac{1}{2} \cos \frac{y+y}{2}=\cos y \text { (Ans.) } \\ \end{array} limx→yx−ysinx−siny=limx→yx−y2sin2x−ycos2x+y=2.limx→y2x−ysin2x−y×21×limx→ycos2x+y=2×1×21cos2y+y=cosy (Ans.)
Evaluate the following limits.
limx→02−x−2+xx\displaystyle\lim_{x\rightarrow 0}\dfrac{\sqrt{2-x}-\sqrt{2+x}}{x}x→0limx2−x−2+x.
limx→1(xx−1−1logx) \lim_{x \rightarrow 1} \left ( \frac{x}{x - 1} - \frac{1}{\log{x}} \right ) limx→1(x−1x−logx1) এর মান কত ?
If the function f(x)=(1−x)tanπx2f(x) = (1 - x)\tan \dfrac{{\pi x}}{2}f(x)=(1−x)tan2πx is continuous at x=1x = 1x=1 ,then f(1)=f(1)=f(1)=
limx→0xtan2x−2xtanx(1−cos2x)2 \displaystyle \lim _{x \rightarrow 0} \dfrac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^{2}} x→0lim(1−cos2x)2xtan2x−2xtanx is equal to
