$\begin{array}{l} The\,series\,\sum\limits_{}^{} {a_n b_n } \,\,\,converges\,\,if\,\,a_{n + 1} \le \,a_{n\,} \,for\,\,n\, \ge \,k,\,\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = 0,\,\,and\, \\\ |b_k + \,b_{k + 1} + ... + b_n | - - - - M,\,\,for\,\,some\,\,constant\,M. \\\ \end{array}$

• A) $\ge \,$
• B) $None\,of\,these$
• C) $\, \le$
• D) $<$

Generalized Mean Value Theorem says, If $f$ and $g$ are continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ then

• A) $$[g(b)-g(a)]=[f(b)-f(a)]g'(c)$$ for some $c$ in $(a,b).$
• B) $$[g(b)-g(a)]f'(c)=[f(b)-f(a)]g'(c)$$ for some $c$ in $(a,b).$
• C) $g'(x)=0$ for all $x$ in $(a,b)$.
• D) $g'(x)=f'(x)$ for all $x$ in $(a,b)$.

$$The\,\,\,function\,\,g\left( x \right) = x^2 \,\,is\, - - - - - \,on\,\,[0,\infty )\,.\,$$

• A) $$inrea\sin g$$
• B) $$decrea\sin g\,\,$$
• C)
• D)

If f(x) = |x| and g(x) = |5x – 2|. Then, fog =​

• A) |f(x)|
• B) f(x)
• C) f^2(x)
• D) |g(x)|

$\begin{array}{l} If\,\,\,\sum\limits_{n = 1}^\infty {b_n } \,\,\,is\,\,rearrangement\,\,of\,\,an\,absolutely\,convergernt\,series \\\ \sum\limits_{n = 1}^\infty {a_n } ,\,\,then\,\,\sum\limits_{n = 1}^\infty {b_n } \,\,also\,\, - - - - - - \,\,absolutely,\,and\,\,to\,the\,\,same\,sum. \\\ \end{array}$

• A) $diverges$
• B) $converges$
• C)
• D)

The derivative of the function ${x^n}$ is

• A) $n{x^n}.$
• B) $n{x^{n - 1}}.$
• C) $\frac{n}{{{x^{n - 1}}}}.$
• D) ${x^{n - 1}}.$

The function $g(x) = {x^2}$ is

• A) having negative values on $[0,\infty ).$
• B) decreasing on $[0,\infty ).$
• C) increasing on $[0,\infty ).$
• D) undefined on $[0,\infty ).$

The value $1+$ $\lim_{x\rightarrow 0^{-}}\frac{\left\vert x\right\vert }{x}$

• A) 1
• B) 3
• C) 0
• D) 2

If $\lim_{x\rightarrow 0}f(x)=20\ $and $\lim_{x\rightarrow 0}g(x)=-5,$ then $\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}$ is

• A) -4
• B) 0
• C) -5
• D) 20

For the function defined as \[f\left( x \right){\text{ }} = {\text{ }}cx,\] for every $\epsilon > 0$ the formal definition ensures

• A) $|f(x) - c{x_0}| < ,\qquad |x - {x_0}| < \delta ,\,\,\,{\text{where}}\,\,\,0 < \delta < \epsilon.$
• B) $|f(x) - c{x_0}| < ,\qquad |x - {x_0}| < \delta ,$ where $ -\epsilon < \delta < \epsilon.$
• C) $|f(x) - c{x_0}| < ,\qquad |x - {x_0}| < \delta ,$ where $0 < \delta < \epsilon/|c|.$
• D) $|f(x) - c{x_0}| < ,\qquad |x - {x_0}| < \delta ,$ where $0 < \delta < \epsilon.$

For the function $$f(x)=x\sin \frac{1}{ x},\quad x\ne0,$$ which statement is true

• A) $\lim_{x\rightarrow 0} f(x) = 0$.
• B) The function is not defined at $x=0$.
• C) $\lim_{x\rightarrow 0} f(x) = 1$.
• D) $\lim_{x\rightarrow 1/\pi} f(x) = 0$.

Which of the following functions is always continuous on its domain?

• A) Piecewise functions.
• B) Polynomial functions.
• C) Rational functions.
• D) Step functions.

If $f(x)=2-\frac{1}{x^{2}}, $then $\lim_{x\rightarrow \infty }f(x)= ----$

• A) 2
• B) 1
• C) 3
• D) 0

A function f is continuous at \(x=c\), if

• A) \(​\lim _{x \to c }f(x)\) exists but does not equal \(f(c).\)
• B) \(f(x)\) is not defined at \(x=c.\)
• C) \(​\lim _{x \to c }f(x)\) exists and equals \(f(c).\)
• D) \(​\lim _{x \to c }f(x)\) does not exist.

For the function $f(x)=x\sin \frac{1}{x},\ \ \ x\neq 0,$\ which statement is true

• A) $\lim_{x\rightarrow 0}f(x)=0\ $
• B) $\lim_{x\rightarrow \frac{1}{x}}f(x)=0\ $
• C) The function is not defined at $x=0$
• D) $\lim_{x\rightarrow 0}f(x)=1\ $

Which of the following best describes a removable discontinuity?

• A) A point where the limit exists but is not equal to the function’s value.
• B) A point where the function has a jump.
• C) A point where the limit does not exist due to infinity.
• D) A point where the function is continuous.

$Consider\,\,the\,\,function\,\,f(x) = \,e^x ;\,\,Find\,\,the\,\,range\,\,of\,\,f\,\,is\,\, - - - - - .$

• A) $none\,\,of\,\,these$
• B) $[0,\infty )\,$
• C) $(0,\infty )\,$
• D) $[ - 1,1]$

If $f(x)=3-\frac{1}{x^{2}},\ \ $then \ $\lim_{x\rightarrow \infty }f(x)=\ \ ----$

• A) 1
• B) 0
• C) 2
• D) 3

For the function f(x)=x\sin \frac{1}{x},\ \ \ x\neq 0,\ which statement is true

• A) \lim_{x\rightarrow 0}f(x)=0\",\lim_{x\rightarrow 0}f(x)=1\"
• B) \lim_{x\rightarrow \frac{1}{x}}f(x)=0\",The function is not defined at x=0"
• C)
• D)

For the function f(x)=x\sin \frac{1}{ x},\quad x\ne0, which statement is true

• A) The function is not defined at x=0
• B) \lim_{x\rightarrow 0} f(x) = 0
• C) \lim_{x\rightarrow 1/\pi} f(x) = 0
• D) \lim_{x\rightarrow 0} f(x) = 1