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

• A) 0
• B) 3
• C) 2
• D) 1

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

• A) 2
• B) 1
• C) 3
• D) 0

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

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

• A) 1
• B) 3
• C) 2
• D) 0

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

• A) 2
• B) 1
• C) 0
• D) 3

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.

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

• A) 1
• B) 0
• C) 2
• D) 3

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

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

If a function f is differentiable at x=c, then f is

• A) Sometimes continuous at x=c.
• B) Never continuous at x=c.
• C) Differentiability at x=c does not imply continuity at x=c.
• D) Always continuous at x=c.

If a function \(f\) is differentiable at \(x=c,\) then \(f\) is

• A) Always continuous at \(x=c.\)
• B) Differentiability at \(x=c\) does not imply continuity at \(x=c.\)
• C) Sometimes continuous at \(x=c.\)
• D) Never continuous at \(x=c.\)

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) 20
• C) -5
• D) 0

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

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

Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0} \,\,\frac{{\sin x\,\,}}{x} = - - - - .

• A) 1\,
• B) x\,
• C) none\,\,of\,\,these
• D) 0

The function h(x) = - {x^3} is

• A) decreasing on ( - \infty ,\infty ).
• B) increasing on ( - \infty ,\infty ).
• C) positive on ( - \infty ,\infty ).
• D) is undefined on ( - \infty ,\infty ).

Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0^ - } \,\frac{1}{x} = - - - - .

• A) \infty \,
• B) none\,\,of\,\,these
• C) - \infty
• D) 0

Rolle's Theorem says, suppose that f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and f(a) = f(b).

• A) Then f'(c) = 0 for some c in the open interval (a,\infty).
• B) Then f'(c) exists for some c in the open interval (a,b).
• C) Then f'(c) = 0 for some c in the open interval (a,b).
• D) Then f'(c) \neq 0 for some c in the open interval (a,b).

If f is differentiable at a local extreme point x_0\in D_{f}^{0}, then

• A) x_0=~0.
• B) |f'(x_0)|<~0.
• C) f'(x_0)=~0.
• D) f'(x_0) \neq~0.

\begin{array}{l} The\,\,radius\,\,\,of\,\,\,convergence\,\,\,of\,\,\,\,\sum\limits_{}^{} {a_n \,(x - x_n )^n \,\,\,is\,\,\,given\,\,\,by\,\, - - - - = \mathop {\lim }\limits_{n \to \infty } } |\frac{{a_{n + 1} }}{{a_n }}|\, \\\ if\,the\,limit\,exists\,in\,the\,extended\,real\,system. \\\ \end{array}

• A) \frac{1}{R}
• B) R
• C) none\,of\,these
• D) n

The\,series\,\sum\limits_{}^{} {( - 1)^n a_n } \,\,\, - - - - - - \,\,\,if\,\,0 \le a_{n + 1} \le \,a_{n\,} \,and\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = 0

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

Which function has 0/0 form

• A) f(x) = \frac{\sin \pi x}{x-1} at x=1
• B) f(x) = 2\ln x at x=2
• C) f(x) = \cos x + x at x =0
• D) f(x) = \frac{e^{-x}}{x} at x=2