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Evaluate Limit x tends to 0 (tanx-sinx)/x^3|$\lim_{x\rightarrow 0}\frac{\tan x – sinx}{x^3}$

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Evaluate Limit x tends to 0 (tanx-sinx)/x^3|$\lim_{x\rightarrow 0}\frac{\tan x – sinx}{x^3}$ – It is an Important Sum of Limit Chapter of Class 11 of Some Important Math Books Like SN Dey, RD Sharma, NCERT, CBSE, ICSE, WBCHSE, JEE, Etc.

$\lim_{x\rightarrow 0}\frac{\tan x – sinx}{x^3}$
=$ \lim_{x\rightarrow 0}\frac{\frac{\sin x}{\cos x} – sinx}{x^3} $
= $\lim_{x\rightarrow 0}\frac{\sin x\left(\frac{1}{\cos x} – 1\right)}{x^3}$
= $\lim_{x\rightarrow 0}\frac{\sin x(1 – cosx)}{x^3cosx}$
= $\lim_{x\rightarrow 0}(\frac{\sin x}{x}\ldotp \frac{2sin^2\frac{x}{2}}{x^2}\ldotp \frac{1}{\cos x})$
= $\lim_{x\rightarrow 0}\left(\frac{\sin x}{x}\ldotp \frac{sin^2\frac{x}{2}}{2.\frac{x^2}{4}}\ldotp \frac{1}{\cos x}\right)$
= $\lim_{x\rightarrow 0}\frac{\sin x}{x}\ldotp \frac{1}{2}\lim_{x\rightarrow 0}\left(\frac{\sin \frac{x}{2}}{\frac{x}{2}}\right)^2\ldotp \lim_{x\rightarrow 0}\frac{1}{\cos x} $
= $\frac{1}{2}\ldotp \lim_{x\rightarrow 0}\frac{\sin x}{x}\ldotp \left(\lim_{x\rightarrow 0}\frac{\sin \frac{x}{2}}{\frac{x}{2}}\right)^2\ldotp \lim_{x\rightarrow 0}\frac{1}{\cos x}$
= $\frac{1}{2}\ldotp 1.1\ldotp 1$
= $\frac{1}{2}$

আরও দেখুন

  1. Evaluate: Limit x tends to zero (cosx-secx)/x^2||$\lim_{x\rightarrow 0}\frac{\cos x – secx}{x^2}$
  2. Evaluate: $\lim_{x\rightarrow 0}\frac{\sin x(1 – cosx)}{x^3} $
  3. Evaluate: $\lim_{x\rightarrow 0}\frac{1 – cos2x}{3tan^2x}$
  4. Limit h Tends to 0 $\lim_{h\rightarrow 0}\frac{\cos ah – cosbh}{h^2}$
  5. Limit x Tends to Zero $\lim_{x\rightarrow 0}\frac{1 – cosx}{\sin^2x}$

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