سلسلة الغاية (طباعة باللاتكس)

$Q26)\lim_{x\to \frac{\pi }{6}}\frac{18sin^{3}x+31sin^{2}x-12sinx-4}{10sin^{3}x+17sin^{2}x-7sinx-2}$

$Q27)\lim_{x\to -5}\frac{x^{2}-25}{arcsin(x+5)}$

$Q28)\lim_{x\to1}\frac{cot\frac{\pi x}{2}}{1-x}$

$Q29)\lim_{x\to 0}\frac{1-cosx \sqrt{cos2x}\sqrt[3]{cos3x}}{x^{2}}$

$Q30)\lim_{x\to1}\frac{1-x}{\pi -2arcsinx}$

$Q31)\lim_{x\to0}\frac{x(1-\sqrt{1-x^{2}})}{\sqrt{1-x^{2}}(racsinx)^{3}}$

$Q32)\lim_{x\to\infty }xcos(\frac{\pi }{4x})sin(\frac{\pi }{4x})$

$Q33)\lim_{x\to1}\frac{1-\sqrt{x}}{(arccosx)^{2}}$

$Q34)\lim_{x\to0}\frac{sin(1+x)-sin(1-x)}{x}$

$Q35)\lim_{x\to0}\frac{1-cos\frac{x}{3}}{1-cos\frac{x}{5}}$

\[Q32)\lim_{x\rightarrow 0}\frac{x\left ( 1-\sqrt{1-x^{2}} \right )}{\sqrt{1-x^{2}}\left ( arcsinx \right )^3}\times \frac{1+\sqrt{1-x^{2}}}{1+\sqrt{1-x^{2}}}\\=\lim_{x\rightarrow 0}\frac{x^{3}}{\left ( arcsinx \right )^{3}}\times \frac{1}{\sqrt{1-x^{2}}\left ( 1+\sqrt{1-x^{2}} \right )}\\=1\times \frac{1}{2}=\frac{1}{2}\]

Q29

$Q36)\lim_{x\to 0}\frac{sec5x-sec3x}{sec3x-secx}$

$Q37)\lim_{x\to\frac{\pi }{6}}\frac{cot^{2}x-3}{cscx-2}$

$Q38)\lim_{x\to\frac{\pi }{4}}\frac{4\sqrt{2}-(cosx+sinx)^{5}}{1-sin2x}$

$Q39)\lim_{x\to0}(tan(\frac{\pi }{4}+x))^{\frac{1}{x}}$

$Q40)\lim_{x\to1}\frac{x^{2}-1}{lnx}$

\[Q36)\lim_{x\rightarrow 0}\frac{sec5x-sec3x}{sec3x-secx}\\=\lim_{x\rightarrow 0}\frac{cos3x-cos5x}{cs5xcos3x}\times \frac{cos3xcosx}{cosx-cos3x}\\=\lim_{x\rightarrow 0}\frac{2sinxsin4x}{cos5x}\times \frac{cosx}{2sinxsin2x}\\=\lim_{x\rightarrow 0}\frac{sin4xcosx}{\mathit{4x}.cos5x}\times \frac{\mathit{4x}.cosx}{sin2x}=2\]

\[\mathbf{Q37)}\lim_{x\rightarrow \frac{\pi }{6}}\frac{cot^{2}x-3}{cscx-2}\\=\lim_{x\rightarrow \frac{\pi }{6}}\frac{csc^{2}x-4}{cscx-2}\\=\lim_{x\rightarrow \frac{\pi }{6}}\frac{(cscx-2)(cscx+2)}{(cscx-2)}=4\]

السلام عليكم حل q40 \[\lim_{x \mapsto 1}\frac{x^2-1}{lnx}=\lim_{x \mapsto 1}\frac{2x}{\frac{1}{x}}=\lim_{x \mapsto1}2x^2=2\]