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Syllabus

limit of x tends to 0

tan 5x divided by sin 3x

limit question

x--> 0

^{2}x))Q50. Let f (xy) = xf (y) + y f (x) for all x, y$\in $R, and f (x) be differentiable in (0 ,$\infty $) then determine f (x).

$53.Supposep\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+.......+{a}_{n}{x}^{n}.If\left|p\left(x\right)\right|\left|\le \right|\left|{e}^{x-1}-1\right|forallx\ge 0.pProvethat\phantom{\rule{0ex}{0ex}}\left|{a}_{1}+2{a}_{2}+....+n{a}_{n}\right|\le 1.$

^{3}+4x^{2}-7/2x^{2}+5x-7x->1

55. Let f(x) = x

^{3}- x^{2}+ x+1 and g(x) $=max\left\{f\left(t\right):0\le t\le x\right\},0\le x\le 1=3-x,1x\le 2.$Discuss the continuity and differentiability of the function g(x) in the interval (0,2).

$56.\mathrm{Let}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{3}-9{\mathrm{x}}^{2}+15\mathrm{x}+6,\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\mathrm{min}\mathrm{f}\left(\mathrm{t}\right):0\le \mathrm{t}\le \mathrm{x}& ,0\le \mathrm{x}\le 6\le \\ \mathrm{x}-18& ,\mathrm{x}6\end{array}\right.,\mathrm{then}\mathrm{draw}\mathrm{the}\mathrm{graph}\mathrm{of}\phantom{\rule{0ex}{0ex}}\mathrm{g}\left(\mathrm{x}\right)\mathrm{and}\mathrm{discuss}\mathrm{the}\mathrm{continutity}\mathrm{and}\mathrm{differentiability}\mathrm{of}\mathrm{g}\left(\mathrm{x}\right).$

X tends to 0

Q). $\underset{x\to 2}{\mathrm{lim}}\frac{\mathrm{log}x-\mathrm{log}2}{x-2}$

Q.6. Find the value of $\underset{n\to \infty}{\mathrm{lim}}\left[\frac{2}{\mathrm{\pi}}\left(n+1\right){\mathrm{cos}}^{-1}\left(\frac{1}{n}\right)-n\right]$.

^{x}-2^{x+1}+1/1- cos x,for x not equal to 0 is continuous at x = 0, find f(0).

ex 7.3 pg135 part2 Q9

Pls solve

$\underset{\mathrm{x}\to 0}{\mathrm{lim}}\frac{{5}^{x}+{3}^{x}-{2}^{x}-1}{x}$

^{3x}- 1 /2xQ. $\underset{\theta \to 0}{\mathrm{lim}}1-\mathrm{cos}\theta \frac{\mathrm{cos}3\theta}{{\theta}^{2}}\mathrm{cos}9\theta $

$\underset{\mathrm{x}\to 0}{\mathrm{lim}}\frac{{15}^{x}-{3}^{x}+1}{{x}^{2}}$

$3)\hspace{0.17em}\underset{\mathrm{x}\to 1}{\mathrm{lim}}\left[\frac{1}{{\mathrm{x}}^{2}+\mathrm{x}-2}-\frac{\mathrm{x}}{{\mathrm{x}}^{3}-1}\right]$

$\left(\mathbf{13}\right)\mathbf{}\underset{x\to 27}{\mathrm{lim}}\mathbf{}\frac{{\mathbf{x}}^{{\displaystyle \raisebox{1ex}{$\mathbf{2}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{9}}{\mathbf{x}\mathbf{}\mathbf{-}\mathbf{}\mathbf{27}}$

Q.52. If $2f\left(x\right)=f\left(xy\right)+f\left(\frac{x}{y}\right)$ for all x, y $\in {R}^{+}$, f (1) = 0 and f' (1) = 1, then find f (e) and f' (2).