What is \lim_{x\rightarrow0}\frac{f(x)}{g(x)} equal to?
Consider the following for the two (02) items that follow : Let the function f(x)=\sin[x], where [x] is the greatest integer function and g(x)=|x|.
- A. -\sin 1
- B. \sin 1
- C. 0
- D. Limit does not exist ✓
Correct Answer: D. Limit does not exist
Explanation
For the left-hand limit (x \to 0^-), g(x) = -x and [x] = -1. So, \lim_{x \to 0^-} \frac{\sin(-1)}{-x} = \infty. Since the left-hand limit is unbounded, the limit does not exist.
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