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When we look at a function's behavior as it approaches a certain point from the right side, we're finding its right-hand limit. This is denoted as \( \lim_{x \to c^+} f(x) \), where "\(+\)" signifies that \(x\) approaches \(c\) from values greater than \(c\). For the function \(f(x) = e^{1/x}\), as \(x\) approaches 0 from the positive side, the term \(1/x\) becomes very large, or tends towards \(+\infty\). As a result, \(e^{1/x}\) also becomes very large because exponential functions increase rapidly when their exponent values increase. Therefore, in this scenario, the right-hand limit of the function as \(x\) approaches 0 is \(\infty\).

This tells us: - As we approach 0 from the right, \(e^{1/x}\) shoots towards infinity. - This behavior makes it clear that the function does not settle towards a finite number on this side.

The left-hand limit looks at how a function behaves when approaching a point from the left side, or from values smaller than \(c\). It's represented as \( \lim_{x \to c^-} f(x) \). For \(f(x) = e^{1/x}\), as \(x\) approaches 0 from the negative side, the expression \(1/x\) becomes largely negative, driving \(e^{1/x}\) towards \(e^{-fty}\). Exponential functions approach zero when their exponents decrease towards \(-\infty\). Thus, the left-hand limit of this function as \(x\) tends to 0 is 0.

Here's what this outcome means: - Approaching 0 from the left, \(e^{1/x}\) gets closer and closer to zero. - Contrasting with the right-hand limit, this shows a drastically different behavior on this side.

Exponential functions, in general, behave distinctively based on their exponential expressions. For \(f(x) = e^{1/x}\), the base \(e\) is a constant approximately equal to 2.718, and it's raised to a power determined by the fractional expression \(1/x\). Its behavior shifts significantly depending on whether \(1/x\) becomes positive or negative:

- **Positive Exponent**: If \(1/x\) is positive, \(e^{1/x}\) escalates sharply, as we observed with the right-hand limit moving to \(+\infty\). - **Negative Exponent**: When \(1/x\) turns negative, \(e^{1/x}\) diminishes, drawing near to zero, exemplified by the left-hand limit.

Key characteristics of exponential functions include: - Rapid growth or decay: They respond swiftly to changes in the exponent. - Asymptotic behavior: As with this function, exponents going towards \(+\infty\) or \(-\infty\) make the function grow indefinitely or shrink towards zero. Understanding this nature helps in analyzing different scenarios of the function's limits.