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

• A) 2
• B) 3
• C) 0
• D) 1

A function f is continuous at $x=c$, if

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

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

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

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

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

Which of the following best describes a removable discontinuity?

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

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

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

On the graph of a function, a removable discontinuity appears as:

• A) A continuous curve.
• B) A vertical asymptote.
• C) A hole in the graph.
• D) A jump discontinuity.

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 functions is always continuous on its domain?

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

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

• A) 1
• B) 3
• C) 0
• D) 2

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 $0 < \delta < \epsilon/|c|.$
• C) $|f(x) - c{x_0}| < ,\qquad |x - {x_0}| < \delta ,$ where $-\epsilon < \delta < \epsilon.$
• 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) 3
• B) 1
• C) 2
• D) 0

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

• A) 0
• B) 3
• C) 1
• 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 two functions f(x) and g(x), the composit function fog(x) is defined as -------

• A) f(g(x))
• B) g(f(x))
• C) None of these
• D) (f-g)(x)

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

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

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

The Mean Value Theorem says, suppose that $f$ is differentiable on $[a,b],$ $f'(a)\ne f'(b),$ and $\mu$ is between $f'(a)$ and $f'(b).$

• A) Then $f'(c)=\mu$ for some $c$ in $(a,b).$
• B) Then $f'(c)\neq\mu$ for some $c$ in $(a,b).$
• C) Then $f'(c)$ may be 0 or $\mu$ for some $c$ in $(a,b).$
• D) Then $f'(c)=0$ for some $c$ in $(a,b).$

Identify an unbounded function on $[0,1]$

• A) $f(x) =x^2+1$
• B) $h(x) = x^3$.
• C) $g(x) = \ln x$.
• D) $v(x) = \sin x$.

$Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0^ - } \,\,\left( {\frac{{\left| x \right|}}{x} + x} \right) = - - - - .$

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