Question

The function $f\left(x\right)=\frac{\cos 3x-\cos 4x}{x\sin 2x}$ for $x\ne 0$, $f\left(0\right)=\frac{7}{4}$ at $x=0$, is

• A. Continuous
• B. discontinuous
• C. left continuous
• D. right continuous

Answer 1

Answer: (A), (C), (D)

The correct options are
A Continuous
C left continuous
D right continuous

From the trigonometric identity,

$cos3x-cos4x=2sin\left(\frac{7x}{2}\right)sin\left(\frac{x}{2}\right)$

$li{m}_{x\to 0}\frac{2sin\left(\frac{7x}{2}\right)sin\left(\frac{x}{2}\right)}{xsin2x}$

$=li{m}_{x\to 0}\frac{sin\left(\frac{7x}{2}\right)}{\frac{7x}{2}}\times \frac{sin\left(\frac{x}{2}\right)}{\frac{x}{2}}\times \frac{x}{x}\times \frac{2x}{sin2x}\times \frac{7}{4}$

Using the property of limits,

$f\left(0\right)=\frac{7}{4}$

Hence, the given function is continuous.