1 Marks Question

Question 11 Mark

Evaluate $\lim _{x \rightarrow 0}\left(e^x-1\right).$

Answer

$\begin{array}{l}\lim _{x \rightarrow 0}\left(e^x-1\right) \\ \lim _{x \rightarrow 0}\left[1+x+\frac{x^2}{\lfloor 2}+\ldots \ldots-1\right]=\lim _{x \rightarrow 0}\left(x+\frac{x^2}{\lfloor{2}}+\ldots \ldots .\right)=0\end{array}$

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Question 21 Mark

$\lim _{x \rightarrow a} \frac{x^m-a^m}{x^n-a^n}, m, n \in N, a>0$

Answer

$\begin{array}{l}\lim _{x \rightarrow a} \frac{x^m-a^m}{x^n-a^n}=\lim _{x \rightarrow a}\left(\frac{x^m-a^m}{x-a}\right)\left(\frac{x-a}{x^n-a^n}\right) \\ =\lim _{x \rightarrow a}\left(\frac{x^m-a^m}{x-a}\right) \div\left(\frac{x^n-a^n}{x-a}\right) \\ =m(a)^{m-1} \div n a^{n-1} \\ =\frac{m\left(a^{m-1}\right)}{n\left(a^{n-1}\right)}=\frac{m}{n}(a)^{m-n}\end{array}$

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Question 31 Mark

Calculate drivative of $\frac{e^x}{x}$ with respect to $x.$

Answer

$\begin{aligned} y & =\frac{e^x}{x} \\ \frac{d y}{d x} & =\frac{x \cdot \frac{d}{d x} e^x-e^x \frac{d}{d x} x}{x^2} \\ & =\frac{x e^x-e^x \cdot 1}{x^2}=\frac{e^x(x-1)}{x^2}\end{aligned}$

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Question 41 Mark

Evaluate $\lim _{x \rightarrow 0} \frac{\tan x}{x}.$

Answer

$\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$
$\because$ It is standard formula.

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Question 51 Mark

Differentiate function $y=\frac{1}{\sin x}$ with respect to $x.$

Answer

$\begin{aligned} y & =\frac{1}{\sin x}=\operatorname{cosec} x \\ \therefore \quad \frac{d y}{d x} & =-\operatorname{cosec} x \cdot \cot x\end{aligned}$

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Question 61 Mark

Write the value of $\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}.$

Answer

$\begin{array}{l}\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta} \\ =\lim _{\theta \rightarrow 0} \frac{2 \sin ^2 \frac{\theta}{2}}{\theta}=\lim _{\theta \rightarrow 0} \frac{\sin \frac{\theta}{2}}{\frac{\theta}{2}} \times \sin \frac{\theta}{2} \\ =\lim _{\theta \rightarrow 0} \frac{\sin \frac{\theta}{2}}{\frac{\theta}{2}} \times \lim _{\theta \rightarrow 0} \sin \frac{\theta}{2}=1 \times 0=0\end{array}$

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Question 71 Mark

Calculate the value of $\lim _{x \rightarrow 0} \frac{(1-x)^n-1}{x}.$

Answer

$\begin{array}{l}\text { Given }: \lim _{x \rightarrow 0} \frac{(1-x)^n-1}{x} \\ =\lim _{y \rightarrow 1} \frac{y^n-1}{y-1} \\ \text { when } y=1-x \text { if } x \rightarrow 0, y=1-x \Rightarrow y \rightarrow 1 \\ =-n(1)^{n-1}=-n\end{array}$

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Question 81 Mark

Evaluate $\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}.$

Answer

$\begin{array}{l}\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}=\lim _{x \rightarrow 0} \frac{2 \sin x \cos x}{x} \\ =2\left[\lim _{x \rightarrow 0} \frac{\sin x}{x}\right]\left(\lim _{x \rightarrow 0} \cos x\right) \\ =2.1 . \cos 0=2.1 . 1=2\end{array}$

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Question 91 Mark

Evaluate $\lim _{x \rightarrow 0}\left(a x^2+b x+c\right).$

Answer

$\begin{array}{l}\lim _{x \rightarrow 0}\left(a x^2+b x+c\right) \\ =0+0+c=c\end{array}$

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Question 101 Mark

Write the standard value of the limit $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^x.$

Answer

$e$.

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Question 111 Mark

Calculate the value of $\lim _{x \rightarrow \infty} e^{\frac{1}{x}}.$

Answer

$\lim _{x \rightarrow \infty} e^{\frac{1}{x}}=e^{\frac{1}{\infty}}=e^0=1$

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Question 121 Mark

Evaluate $\lim _{x \rightarrow 0}(x \cdot \cot x).$

Answer

$\begin{array}{l}\lim _{x \rightarrow 0} x \cdot \cot x \\ =\lim _{x \rightarrow 0} \frac{x}{\tan x}=\lim _{x \rightarrow 0} \frac{1}{\frac{\tan x}{x}}=\frac{1}{1}=1\end{array}$

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Question 131 Mark

$\lim _{x \rightarrow 2} \frac{x^2-3 x+2}{x^2+x-6}$

Answer

$\begin{array}{l}\lim _{x \rightarrow 2} \frac{x^2-3 x+2}{x^2+x-6}=\lim _{x \rightarrow 2} \frac{(x-2)(x-1)}{(x-2)(x+3)}=\lim _{x \rightarrow 2} \frac{(x-1)}{(x+3)} \\ =\frac{2-1}{2+3}=\frac{1}{5}\end{array}$

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