Substitution Method to Remove Indeterminate Form

Q.

Which of the following is the expansion of tan x?

1. 

2. 

3. 

4. 

5. 

Q.

Find the domain and the range of the real function, $f\left(x\right)=\frac{x^2+1}{x^2-1}$

Q.

Find the angle between z and iz in degrees.

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Q. The value of $\underset{x\to \infty }{\lim }\frac{{\left(x+1\right)}^{10}+{\left(x+2\right)}^{10}+...+{\left(x+100\right)}^{10}}{x^{10}+{10}^{10}}$ is

(a) 10

(b) 100

(c) 10¹⁰

(d) none of these

Q. The value of $\underset{x\to \infty }{\lim }\frac{\sqrt{1+x^4}+\left(1+x^2\right)}{x^2}$ is

(a) −1
(b) 1
(c) 2
(d) none of these

Q. $\underset{n\to \infty }{\lim }\frac{1^2+2^2+3^2+...+n^2}{n^3}$ is equal to

(a) 1
(b) 1/2
(c) 1/3
(d) 0

Q. The value of the limit $\underset{x\to \infty }{\lim }\left(\sqrt{x+3}-\sqrt{x}\right\}\sqrt{x+1}$ is

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

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

Q.

Let $f:R\to R:f\left(x\right)=x+2$ and $g:R-\left\{2\right\}\to R:g\left(x\right)=\frac{x^2-4}{x-2}$
Show that $f\ne g$. Re-define f and g such that f = g

Q. The value of $\underset{n\to \infty }{\lim }\left\{\frac{1+2+3+...+n}{n+2}-\frac{n}{2}\right\}$ is

(a) 1/2

(b) 1

(c) −1

(d) −1/2

Q. Let $f\left(x\right)=\frac{\mathrm{ax}+b}{x+2},\underset{x\to 0}{\lim }f\left(x\right)=2\mathrm{and}\underset{x\to \infty }{\lim }f\left(x\right)=1.$ The value of f(-1) is

Q. Let f(x) be a function defined by $f\left(x\right)=\left\{\begin{array}{ll}\frac{3x}{\left|x\right|+2x},& x\ne 0\\ 0,& x=0\end{array}\right..$
Show that $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist.

Q. Solve the equation $tanx+tan2x+\sqrt{3}tanxtan2x=\sqrt{3}$.
Or
Prove that $sin{18}^{\circ }=\frac{\sqrt{5}-1}{4}$.

Q. Let $a,b,\in R,b\ne 0.$ Define a function
$f\left(x\right)=\{\begin{array}{cc}a\sin \frac{\pi }{2}(x-1),& \text{for}x\le 0\\ \frac{\tan 2x-\sin 2x}{b{x}^3},& \text{for}x>0\end{array}.$
If $f$ is continuous at $x=0,$ then $10-ab$ is equal to

Q. $\underset{n\to \infty }{\lim }\frac{1-2+3-4+5-6+....+\left(2n-1\right)-2n}{\sqrt{n^2+1}+\sqrt{n^2-1}}$ is equal to

(a) $\frac{1}{2}$

(b) $-\frac{1}{2}$

(c) 1

(d) −1

Q. $\underset{x\to 0}{lim}\frac{x\cot \left(4x\right)}{{\sin }^{2}x{\cot }^2\left(2x\right)}$ is equal to :

1. $2$
2. $0$
3. $1$
4. $4$

Q.

Find the value of $\underset{x\to \infty }{lim}\frac{8-\frac{3}{x}+\frac{5}{x^2}}{4+\frac{2}{x}+\frac{12}{x^2}}$

Q. $\underset{n\to \infty }{\lim }\left[\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}\right]$

Q. $\underset{n\to \infty }{\lim }\left[\frac{1^3+2^3+....n^3}{n^4}\right]$

Q. Find $\underset{x\to 1^{+}}{\lim }\left(\frac{1}{x-1}\right).$

Q. If a = 1 + b + b² + b³ + ... to ∞, then write b in terms of a.