Question

Which of the following statements is/are TRUE?

• A. If f has domain $[0, \infty)$ and has no horizontal asymptotes then $\underset{ x \rightarrow \infty} {\text{Lim}} f( x )=\infty$ or $\underset{ x \rightarrow \infty} {\text{Lim}} f( x )=-\infty$.
• B. If $f$ is continuous on $[-1,1]$ and $f(-1)=4$ and $f(1)=3$ then there exist a number $r$ such that $|r|< 1$ and $f(r)=\pi$.
• C. $\underset{ x \rightarrow \infty} {\text{Lim}}\arcsin \left(\frac{x+1}{x}\right)$ does not exist.
• D. For all values of $m \in R$ the line $y-m x+m-1=0$ cuts the circle $x^2+y^2-2 x-2 y+1=0$ orthogonally.

Answer 1

Answer: (B), (C), (D)

(A) False, e.g. $f(x)=\sin \sqrt{x}$
(B) True, from IVT
(C) True as $\underset{ x \rightarrow \infty}{\text{Lim}} \sin ^{-1}\left(1+\frac{1}{ x }\right)=\sin ^{-1}$ (a quantity greater than one) $\Rightarrow$ not defined
(D) True, as the line passes through the centre of the circle.