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The graph of \(g\) has a discontinuity at \(x=2\), marked by the vertical dashed asymptote. Determine from the graph whether \(\lim_{x\to2}g(x)\) exists. Justify your answer by describing the function values near the discontinuity.
Hints
- Examine the graph separately from the left and from the right of \(x=2\).
- One-sided limits may be infinite.
- A two-sided limit exists only when the two one-sided limits agree.
Solution
1. As \(x\to2^-\), the function values decrease without bound, so the left-hand limit is \(-\infty\).
2. As \(x\to2^+\), the function values increase without bound, so the right-hand limit is \(+\infty\).
3. A two-sided limit requires the one-sided limits to agree. Since they approach infinities with opposite signs, \(\lim_{x\to2}g(x)\) does not exist.
Answer
The limit does not exist because \(g(x)\to-\infty\) as \(x\to2^-\), while \(g(x)\to+\infty\) as \(x\to2^+\).
