Expansions to Remove Indeterminate Form

Q.

Find the value of other five trigonometric functions if tan x = $\frac{-5}{12}$, and x lies in second quadrant.

Q.

Find the value of other five trigonometric functions if sin x = $\frac{3}{5}$, and x lies in second quadrant.

Q.

You are given $cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}......;$

$sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}......$

$tanx=x+\frac{x^3}{3}+\frac{2.x^5}{15}......$

Find the value of $\underset{x\to 0}{lim}\frac{xcosx+sinx}{x^2+tanx}$

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Q. $\underset{x\to 0}{lim}\frac{1-cos3x}{x(3^x-1)}=$

1. 
2. 
3. 
4. 

Q.

${\int }_0^{\frac{\mathrm{\pi }}{4}}\log \left(1+tanx\right)dx=$

1. $\frac{\mathrm{\pi }}{8}{\log }_{e}2$

2. $\frac{\mathrm{\pi }}{4}{\log }_{2}e$

3. $\frac{\mathrm{\pi }}{4}{\log }_{e}2$

4. $\frac{\mathrm{\pi }}{8}{\log }_e\frac{1}{2}$

Q. $\underset{x\to 0}{lim}\frac{\sin x-x+\frac{x^3}{6}}{x^5}$ is equal to

1. 
2. 
3. 
4. 

Q.

You are given $cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}......;$

$sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}......$

$tanx=x+\frac{x^3}{3}+\frac{2.x^5}{15}......$

Find the value of $\underset{x\to 0}{lim}\frac{xcosx+sinx}{x^2+tanx}$

Q.

Evaluate ${lim}_{x\to 3}\frac{x^2-9}{x-3}$

Q.

Evaluate$\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{(\sqrt{2}-\sqrt{1+\cos x})}$:

1. $\sqrt{2}$

2. $2$

3. $4$

4. $4\sqrt{2}$

Q.

If $y=(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...\infty )$show that$\frac{dy}{dx}=y.$

Q.

How many of the following trigonometric ratios are correct?

(1)Sin (${15}^{\circ }$) = $\frac{\sqrt{3}+1}{2\sqrt{2}}$

(2)Cos (${105}^{\circ }$) = $\frac{1-\sqrt{3}}{2\sqrt{2}}$

(3)Sin ${75}^{\circ }$ = $\frac{\sqrt{3}-1}{2\sqrt{2}}$

(4)Tan ${15}^{\circ }$ = $2-\sqrt{3}$

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Q. $\underset{x\to 0}{lim}\frac{\cos \left(\tan x\right)-\cos x}{x^4}$ is equal to

1. $\frac{1}{6}$
2. $\frac{-1}{3}$
3. $\frac{1}{2}$
4. $1$

Q.

You are given
$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}......$;
$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}......$;
$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}......$
Then the value of
$\underset{x\to 0}{lim}\frac{x\cos x+\sin x}{x^2+\tan x}$ is

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Q. $\underset{x\to 0}{lim}{\left(\frac{1+2x}{1+3x}\right)}^{\frac{1}{x^2}}.e^{\frac{1}{x}}=$

1. $e^{\frac{5}{2}}$
2. $e^2$
3. 2
4. 1

Q.

The number of ordered pair (a, x) satisfying the equation $se{c}^2$(a + 2)x + $a^2$ - 1 = 0; -$\pi$ < x < $\pi$ is

1. 2

2. 1

3. 3

4. Infinite

Q. Let $\alpha ,\beta \in R$ be such that $\underset{x\to 0}{lim}\frac{x^2\sin \left(\beta x\right)}{\alpha x-\sin x}=1.$ Then $6(\alpha +\beta )$ equals

Q. $\underset{x\to \frac{\pi }{2}}{lim}\frac{\cot x-\cos x}{(\pi -2x{)}^3}$ is equal to

1. 
2. 
3. 
4. 

Q. If $\frac{sin(x-\alpha )}{sin(x-\beta )}=\frac{a}{b},\frac{cos(x-\alpha )}{cos(x-\beta )}=\frac{A}{B}$ and aB + bA $\ne$ 0, then prove that -
$cos(\alpha -\beta )=\frac{aA+bB}{aB+bA}.$

Q. $\underset{x\to 0}{lim}\frac{\sin x-x+\frac{x^3}{6}}{x^5}$ is equal to

1. $\frac{1}{120}$
2. $\frac{1}{160}$
3. $\frac{1}{2}$
4. $0$

Q. Let $f\left(x\right)=1+e^{\ln \left(\ln x\right)}\cdot \ln (k^2+25)$ and $g\left(x\right)=\frac{1}{|x|-1}$. If $\underset{x\to 1}{lim}f(x{)}^{g\left(x\right)}=k(2{\sin }^2\alpha +3\cos \beta +5)$, $k>0$ and $\alpha ,\beta \in R$, then which of the following statement(s) is (are) CORRECT?

1. $k=5$
2. $\frac{{\sin }^{10}\alpha +{\cos }^5\beta }{{\sin }^2\alpha +\cos \beta }=1$
3. ${\cos }^2\beta +{\sin }^4\alpha =2$
4. ${\sin }^2\alpha >\cos \beta$

Q. $\underset{x\to 0}{lim}\frac{e^{\sin x}-\sin x-1}{x^2}$ is equal to

1. $\frac{1}{3}$
2. $1$
3. $e$
4. $\frac{1}{2}$

Q. 17. imC2x-1x→0 cosx-1, lim

Q. tan 2x22· 1-1

Q. ${lim}_{x\to 1}(\frac{2}{1-x^2}+\frac{1}{x-1})=$___

1. 1
2. 0
3. $\infty$
4. $\frac{1}{2}$

Q. Solve:
$\underset{x\to 0}{lim}\frac{e^{\sin 3x}-1}{\log \left(1+\tan 2x\right)}=\frac{3}{2}$

Q.

The value of $\underset{x\to 0}{lim}\frac{(1+x{)}^{1/x}-e+\frac{1}{2}ex}{x^2}$ is

1. $\frac{11}{24}e$

2. $\frac{11e}{24}$

3. $\frac{e}{24}$

4. $\frac{e}{11}$

Q.

$\underset{x\to 0}{lim}\left[\frac{lncosx}{\sqrt[4]{41+x^2}-1}\right]\text{is equal to}$

1. 2

2. -2

3. 1

4. -1