Question 11 Mark
Find the value of $\operatorname{Lim}_{x \rightarrow-3} \sqrt[3]{2-2 x}$.
View full question & answer→Question 21 Mark
Find the value of $\lim _{x \rightarrow 5}(3 x+5)$.
Answer$\lim _{x \rightarrow 5}(3 x+5)=3(5)+5=20$
View full question & answer→Question 31 Mark
If $|x + 4| < 0.01 = (K, -3.96)$ then find the value of $K.$
AnswerComparing $|x + 4| < 0.04$ with $|x - a| < δ;$
we get $a = -4$ and $δ = 0.04.$
Now, $N(a, δ) = (a - δ; a + δ)$
$\therefore N(-4, 0.04) = (-4 - 0.04; -4 + 0.04)$
$= (-4.04; -3.96)$
Hence, $(k, -3.96) = (-4.04; -3.96)$
$\therefore k = -4.04$
View full question & answer→Question 41 Mark
If $N (a, 0.2) = |x – 7|< b$ then find the value of $a.$
AnswerComparing $N(a, 0.2) = |x - 7| < b$ with $N(a, δ)$
$=|x - a| < δ;$ we get $a = 7$ and $b = 0.2$
View full question & answer→Question 51 Mark
Express $N (50, 0.8)$ in modulus form.
AnswerComparing $N (50, 0.8)$ with $N (a, δ);$
we get $a = 50$ and $δ = 0.8$
modulus form: $|x – a|<δ$
Putting $a=50$ and $δ = 0.8$
$N(50, 0.8) = Ix - 50| < 0.8$
View full question & answer→Question 61 Mark
Express $|2 x|<\frac{1}{2}$ in interval form.
Answer$|2 x|<\frac{1}{2}$ in interval form is expressed by $\left(-\frac{1}{4}, \frac{1}{4}\right)$
View full question & answer→Question 71 Mark
Express $|x-10|<\frac{1}{10}$ in neighbourhood form.
Answer$|x-10|<\frac{1}{10}$ in neighbourhood form is expressed by $N\left(10, \frac{1}{10}\right)$.
View full question & answer→Question 81 Mark
Express $0.001$ neighbourhood of $-5$ in modulus form.
AnswerComparing $0.001$ neighbourhood of $-5$ with $δ$ neighbourhood of a $N(a, δ);$
we get $a = -5$ and $δ = 0.001.$
Modulus form $: |x - a| < δ$
Putting $a = -5$ and $δ = 0.001.$
$0. 001$ neighbourhood of $-5 = |x + 5| < 0.001$
View full question & answer→Question 91 Mark
If $\lim _{x \rightarrow 3} \frac{2}{3 x+k}=\frac{1}{7}$, then find the values of $k$.
View full question & answer→Question 101 Mark
If $\lim _{x \rightarrow-1} 4 x+k=6$, then find the value $k$.
Answer$\lim _{x \rightarrow-1} 4 x+k=6$
$ =6=4(-1)+k=6$
$ \therefore k=6+4=10$
View full question & answer→Question 111 Mark
Find the value of $\lim _{x \rightarrow-a} \frac{x^m+a^m}{x+a}$ where $m$ is an odd number.
Answer$\begin{aligned} & \lim _{x \rightarrow-a} \frac{x^m+a^m}{x+a} \\ & =\lim _{x \rightarrow-a} \frac{x^m-\left(-a^m\right)}{x-(-a)} \\ & =m(-a)^{m-1} \quad\left(\because \lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n \cdot a^{n-1}\right) \\ & =m a^{m-1}(\because m \text { is odd, so } m-1 \text { is even. })\end{aligned}$
View full question & answer→Question 121 Mark
Find the value of $\lim _{x \rightarrow 2} \frac{x^5-32}{x-2}$
Answer$\begin{aligned} & \lim _{x \rightarrow 2} \frac{x^5-32}{x-2} \\ & =\lim _{x \rightarrow 2} \frac{x^5-2^5}{x-2} \quad \\ & =5 \times 2^{5-1}\left(\because \lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n \cdot a^{n-1}\right) \\ & =5 \times 2^4=5 \times 16=80\end{aligned}$
View full question & answer→Question 131 Mark
Express $0.09$ neighbourhood of $0$ in interval form.
AnswerComparing $0.0-9$ neighbourhood of $0$ with $δ$ neighbourhood of a $N(a, δ);$
we get $a = 0$ and $δ = 0.09.$
Interval form$ : (a - δ ; a + δ)$
Putting $a = 0$ and $δ = 0.09$
$0.09$ neighbourhood of $0 = (0 - 0.09; 0 + 0.09),$
$= (-0.09; 0.09)$
View full question & answer→Question 141 Mark
Express the following in interval and neighbourhood form: $|x+3|<0.15$
AnswerIn Interval form: $\{-3.15,-2.85\}$In Neighbourhood form: $N(-3,0.15)$
View full question & answer→Question 151 Mark
Express the following in interval and neighbourhood form: $|x| < \frac{1}{3}$
AnswerIn Interval form: $-\frac{1}{3},\frac{1}{3}$ In Neighbourhood form: $ N(0, \frac{1}{3})$
View full question & answer→Question 161 Mark
Express the following in interval and neighbourhood form: $|x + 5 | < 0.1$
AnswerIn Interval form: $(-5.1, -4.9)$ In Neighbourhood form: $( - 5, 0.1)$
View full question & answer→Question 171 Mark
Express the following in interval and neighbourhood form $:|x – 2|< 0.01$
AnswerIn Interval form $: (1.99, 2.01)$ In Neighbourhood form $:N(2, 0.01)$
View full question & answer→Question 181 Mark
Express the following in modulus and interval form $: 0.001$ neighbourhood of $- 1$
AnswerIn Modulus form $: |x+1 |< 0.001$ In Interval form $: (- 1.001, - 0.999)$
View full question & answer→Question 191 Mark
Express the following in modulus and interval form $: 0.05$ neighbourhood of $0$
AnswerIn Modulus form $: |x|$ In Interval form $:\ =(- 0.05, 0.05)$
View full question & answer→Question 201 Mark
Express the following in modulus and interval form $: 0.02$ neighbourhood of $2$
AnswerIn Modulus form $: |x - 2|< 0.02$ In Interval form $: (1.98, 2.02)$
View full question & answer→Question 211 Mark
Express the following in modulus and interval form $: 0.4$ neighbourhood of $4$
AnswerIn Modulus form $: |x - 4|$ In Interval form $: (3.6, 4.4)$
View full question & answer→Question 221 Mark
If $N(k, 5)=(b, 7)$, find $k$ and $b$.
View full question & answer→Question 231 Mark
If $(a-\delta, a+\delta)=(-3,7)$, find a and $\delta$.
View full question & answer→Question 241 Mark
Find the value of $\lim _{h \rightarrow 0} \frac{f(x+h)}{5}$, where $f(x)=3 k$
View full question & answer→Question 251 Mark
Obtain limit of $\lim _{x \rightarrow-1} \frac{x^{3}-3}{x-1}$
View full question & answer→Question 261 Mark
Find the value of $\lim _{x \rightarrow-4} \frac{x^{2}-16}{4}$
View full question & answer→Question 271 Mark
State the value of $\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$
View full question & answer→Question 281 Mark
What is the value of $\lim _{x \rightarrow 0} \frac{1}{x}$
AnswerThe value of $\lim _{x \rightarrow 0} \frac{1}{x}$ does not exist.
View full question & answer→Question 291 Mark
Find the value of $\lim _{x \rightarrow-\frac{2}{3}} \frac{x^{2}+1}{x^{2}-1}$
View full question & answer→Question 301 Mark
If $\lim _{x \rightarrow a} f(x)=l$ and $\lim _{x \rightarrow a} g(x)=m$ write the condition for $\lim _{x \rightarrow a}\left[\frac{f(x)}{g(x)}\right]$
View full question & answer→Question 311 Mark
What is the meaning of $x \rightarrow-1$ ?
Answer$x \rightarrow-1$ means $x$ tends to $-1$, but $x \neq-1$
View full question & answer→Question 321 Mark
Find the value of $\lim _{x \rightarrow 0} \frac{x^{3}+2 x^{2}+x}{3 x^{2}+x}$
View full question & answer→Question 331 Mark
Find the value of $\lim _{x \rightarrow k} \frac{2 x}{k}$
View full question & answer→Question 341 Mark
$f(x)=\frac{3 x+1}{x+1}$ and $g(x)=\frac{(3 x-1)}{\left(x^{2}+1\right)}$ If $x \rightarrow 2$ find $\lim _{x \rightarrow 2}[f(x)-g(x)]$
View full question & answer→Question 351 Mark
When does the limit of function not exist?
AnswerWhen $x \rightarrow a$, then if $f ( x )$ does not tend to a definite value, then the limit of function does not exist.
View full question & answer→Question 361 Mark
If the value of $f(x)$ is indeterminate when $x \rightarrow-3$, state the common factor of $f(x)$
AnswerIf the value of $f(x)$ is indeterminate when $x \rightarrow-3$, the common factor of $f(x)$ is $(x+$ 3).
View full question & answer→Question 371 Mark
When is the value of $f(x)$ indeterminate?
AnswerWhen $x \rightarrow a$ then by putting $x=a$ in $f(x)$, if we get $f(x)=\frac{0}{0}$ the value of function is indeterminant.
View full question & answer→Question 381 Mark
When $x \rightarrow 0$, which values of $x$ are taken from left hand side?
AnswerWhen $x \rightarrow 0$, values of $x$ from left hand side are $-0.1,-0.01,-0.001,-0.0001$
View full question & answer→Question 391 Mark
What is $x \rightarrow a ?$
Answer$x \rightarrow a$ means values of $x$ is closer and closer to a but $x \neq a .$
View full question & answer→Question 401 Mark
What is modulus form of interval form $(a-\delta, a+\delta) ?$
AnswerThe modulus form of interval form $(a-\delta . a+\delta)$ is $|x-a|<8 .$
View full question & answer→Question 411 Mark
Define punctured $\delta$ neighbourhood of a.
AnswerIf a is a real number and $\delta$ is a nonnegative number then $(a-\delta, a+\delta)-\{a\}$ is called punctured $\delta$ neighbourhood of a. It is denoted by $N *(a, \delta)$.
View full question & answer→Question 421 Mark
What is called $N *(a, \delta) ?$
Answer$N *(a, \delta)$ is called punctured $\delta$ neighbourhood of a.
View full question & answer→Question 431 Mark
State the modulus form of $N(a, \delta)$.
AnswerThe modulus form of $N(a, \delta)$ is $|x-a|<\delta$.
View full question & answer→Question 441 Mark
What is a neighbourhood of a called?
AnswerIf ' $a$ ' is a real number, then an open interval containing $a , a \in R$ Is called a neighbourhood of a.
View full question & answer→Question 451 Mark
State the meaning of $|x-a|<\delta$
Answer$|x-a|<\delta \text { means } x \in(a-\delta, a+\delta) .$
View full question & answer→Question 461 Mark
How is the modulus of a real number?
AnswerThe modulus of a real number is always positive.
View full question & answer→Question 471 Mark
Give definition of closed interval.
Answer$a$ and $b$ are real numbers and $a < b$. Set of all numbers between $a$ and $b$ including $a$ and $b$ is called a close interval. It is denoted by $[a, b]$.
View full question & answer→Question 481 Mark
State the definition of interval.
AnswerA set of real numbers between any two real numbers is called an interval.
View full question & answer→Question 491 Mark
AnswerA real line is a line where its points are real numbers.
View full question & answer→Question 501 Mark
Which is one method for finding confident approximate value of a function?
AnswerOne method for finding confident approximate value of a function is limit.
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