Maths · Limits, continuity and differentiability
The function = \) at \right. is
The function \( f(x)= \) \( \left\{\begin{array}{l}\frac{\cos 3 x-\cos 4 x}{x^{2}}, \text { for } x \neq 0 \\ \frac{7}{2}, \text { for } x=0\end{array} \) at \right. \( \boldsymbol{x}=\mathbf{0} \) is
- A. right continuous only
- B. discontinuous
- C. left continuous only
- D. continuous
Step-by-step solution
Using the identity cos3x - cos4x = 2 sin(7x/2) sin(x/2), we rewrite f(x) = [2 sin(7x/2) sin(x/2)] / x^2 = (7/2) * [sin(7x/2)/(7x/2)] * [sin(x/2)/(x/2)]. As x→0, sin(ax)/(ax) → 1, so limit = 7/2 = f(0). Hence f is continuous at x=0.
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