Continuity and Differentiability
Continuity Involving Limits
Premium Question
Grade 12
Question:
If $f(x) = \begin{cases} \frac{1 - \cos x}{x^2}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$, then $k =$
(1) $0$
(2) $\frac{1}{2}$
(3) $-\frac{1}{2}$
(4) $1$
Step-by-Step Solution
Key Concept: $\lim_{x \to 0} \frac{1-\cos x}{x^2} = \lim_{x \to 0} \frac{2 \sin^2(x/2)}{x^2} = \frac{2\cdot(1/4)}{1} \cdot \left(\frac{\sin(x/2)}{x/2}\right)^2 \to \frac{1}{2}$. So $k = \frac{1}{2}$.
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Correct Answer: (2)