Which of the following statements is/are TRUE?

Question

Which of the following statements is/are TRUE? (A) If f has domain [0, ∞) and has no horizontal asymptotes then underset x arrow ∞ textLim f( x )=∞ or underset x arrow ∞ textLim f( x )=-∞. (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)=π. (C)  underset x arrow ∞ textLim arcsin ((x+1/x)) does not exist. (D) For all values of m ∈ R the line y-m x+m-1=0 cuts the circle x2+y2-2 x-2 y+1=0 orthogonally.

Tardigrade Admin · Accepted Answer

(A) False, e.g. f(x)= sin √x (B) True, from IVT (C) True as underset x arrow ∞ textLim sin -1(1+(1/ x ))= sin -1 (a quantity greater than one) ⇒ not defined (D) True, as the line passes through the centre of the circle.

Q. Which of the following statements is/are TRUE?

If f has domain $[0, \infty)$ and has no horizontal asymptotes then $x \to \infty Lim f (x) = \infty$ or $x \to \infty Lim f (x) = - \infty$ .

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 $f (r) = π$ .

$x \to \infty Lim arcsin (\frac{x + 1}{x})$ does not exist.

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.

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

Q. Which of the following statements is/are TRUE?

If f has domain [0,∞) and has no horizontal asymptotes then x→∞Lim​f(x)=∞ or x→∞Lim​f(x)=−∞.

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)=π.

x→∞Lim​arcsin(xx+1​) does not exist.

For all values of m∈R the line y−mx+m−1=0 cuts the circle x2+y2−2x−2y+1=0 orthogonally.

(A) False, e.g. f(x)=sinx​ (B) True, from IVT (C) True as x→∞Lim​sin−1(1+x1​)=sin−1 (a quantity greater than one) ⇒ not defined (D) True, as the line passes through the centre of the circle.

If f has domain $[0, \infty)$ and has no horizontal asymptotes then $x \to \infty Lim f (x) = \infty$ or $x \to \infty Lim f (x) = - \infty$ .

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 $f (r) = π$ .

$x \to \infty Lim arcsin (\frac{x + 1}{x})$ does not exist.

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.

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