Maths · Limits, continuity and differentiability
=\left\{\begin{array}{ll}\frac{3 x+4 \tan x}{x} & \text { for } x \neq 0 \\ 7 &
\( \operatorname{Let} \mathbf{f}(\boldsymbol{x})=\left\{\begin{array}{ll}\frac{3 x+4 \tan x}{x} & \text { for } x \neq 0 \\ 7 & \text { for } x=0\end{array}\right. \) \( \operatorname{then} f(x) \) is
- A. continuous at \( x=0 \)
- B. not continuous at \( x=0 \)
- C. not determined at \( x=0 \)
- D. \( L t_{x \rightarrow 0} f(x)=8 \)
Step-by-step solution
Compute limit as x→0: lim_{x→0} (3x + 4 tan x)/x = 3 + 4 lim_{x→0} (tan x/x) = 3 + 4*1 = 7. Since f(0)=7, lim_{x→0} f(x) = f(0), so f is continuous at x=0.
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