Question
lim_(x-> 7^{-}) f(x)=2 and lim_(x-> 7^{+}) f(x)=2, but f(7)=-2. What can you say about lim_(x->7)f(x)?
Ask by Schneider Harrington.
Mar 23,2025 20:18
UpStudy AI Solution
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Answer
The function \( f(x) \) has a removable discontinuity at \( x=7 \) because the limit as \( x \) approaches 7 is 2, but \( f(7) \) is -2. To fix this, we can redefine \( f(7) \) to be 2.
Solution
1. We are given that
\[
\lim_{x\to7} f(x) = 2,
\]
and that
\[
\lim_{x\to7^{+}} f(x)=2,
\]
but the function value is
\[
f(7)=-2.
\]
2. The limit as \(x\to7\) exists and equals \(2\), regardless of the value of the function at \(7\). This means that as \(x\) approaches \(7\) (from both the left and the right), the function values get arbitrarily close to \(2\).
3. However, the fact that \(f(7)=-2\) indicates that the function does not take the limiting value at \(x=7\); that is,
\[
f(7) \neq \lim_{x\to7} f(x).
\]
4. This discrepancy implies a discontinuity at \(x=7\). Specifically, since the limit exists but does not equal the function’s value at that point, the discontinuity is removable.
5. To remove the discontinuity, we could redefine the function at \(x=7\) such that
\[
f(7)=2.
\]
6. In summary, the function \(f(x)\) has a removable discontinuity at \(x=7\) because the limit exists and equals \(2\) while \(f(7)\) is assigned the value \(-2\).
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Simplify this solution Beyond the Answer
In this scenario, we see an interesting feature of limits and continuity! The fact that \(\lim_{x \rightarrow 7} f(x) = 2\) and \(\lim_{x \rightarrow 7^{+}} f(x) = 2\) tells us that as \(x\) approaches 7 from the left or the right, the function \(f(x)\) is approaching 2. However, \(f(7) = -2\) indicates a discontinuity at that point, making \(f(x)\) not continuous at \(x = 7\). This is a classic example that shows limits can exist even when the function itself does not behave nicely at that point!
Understanding such cases helps solidify the concept of pointwise limits in real analysis. In real-world applications, similar behaviors can arise in systems where expected outcomes differ from actual results, such as in economic models or engineering simulations where certain inputs yield surprising outputs. For instance, a material might behave predictably under certain temperatures, but a specific threshold could cause a break in that expected behavior—akin to reaching your limit!
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