Question

If $f(x) = a\cos(bx + c) + d,f(x)$ is finite, then the value of a is

• A. 3
• B. 5
• C. 2
• D. Any real number

Answer 1

Answer: (C)

2

Answer 2

$\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 4 } + a x ^ { 3 } + 3 x ^ { 2 } + b x + 2 } \right)$

$- \sqrt { \left. x ^ { 4 } + 2 x ^ { 3 } - c x ^ { 2 } + 3 x - d \right) }$

= $\lim _ { x \rightarrow \infty } \left\{ \frac { \left( x ^ { 4 } + a x ^ { 3 } + 3 x ^ { 2 } + b x + 2 \right) } { \sqrt { x ^ { 4 } + a x ^ { 3 } + 3 x ^ { 2 } + b x + 2 } } \right.$

$\lim _ { x \rightarrow \infty } \left\{ \frac { ( a - 2 ) x ^ { 3 } + ( 3 + c ) x ^ { 2 } } { \sqrt { x ^ { 4 } + a x ^ { 3 } + 3 x ^ { 2 } + b x + 2 } } \right.$

Clearly, the degree of the polynomial in the numerator is 3 and that of denominator is 2. Therefore, for the limit to be finite, we must have, a – 2 = 0 ⇒ a = 2