Question

Find $\underset{x\to 0^{+}}{lim}\frac{e^{-1/x}}{x}$. [Hint: Substitute $z=\frac{1}{x}\to +\mathrm{\infty }$. Then use the rule.]

Short answer

The limit is 0.

Step-by-step solution

Understand the Limit

The exercise asks us to find the limit as $x$ approaches 0 from the positive side of the function $\frac{e^{-1/x}}{x}$. To handle this, we'll use a substitution to simplify it.

Perform a Substitution

Set $z=\frac{1}{x}$. As $x$ approaches 0 from the positive side, $z$ approaches infinity since $x$ is positive and very small. This transforms our limit problem to finding $\underset{z\to \mathrm{\infty }}{lim}\frac{e^{-z}}{1/z}$.

Simplify the Expression Using Substitution

The expression $\frac{e^{-1/x}}{x}$ becomes $z\cdot e^{-z}$ after substitution since $\frac{1}{x}=z$. Our limit now reads $\underset{z\to \mathrm{\infty }}{lim}z\cdot e^{-z}$.

Apply L'Hôpital's Rule

The limit $\underset{z\to \mathrm{\infty }}{lim}z\cdot e^{-z}$ is in an indeterminate form ($\mathrm{\infty }\cdot 0$). Rewriting it as $\underset{z\to \mathrm{\infty }}{lim}\frac{z}{e^z}$, L'Hôpital's Rule can be applied. Take derivatives: the derivative of the numerator is 1 and the denominator is $e^z$.

Calculate using L'Hôpital's Rule

After applying L'Hôpital's Rule once, we get:$\underset{z\to \mathrm{\infty }}{lim}\frac{1}{e^z}$. As $z$ tends to infinity, $e^z$ tends to infinity, so $\frac{1}{e^z}$ tends to 0.

Conclude the Limit Calculation

Thus, $\underset{z\to \mathrm{\infty }}{lim}z\cdot e^{-z}=0$. Substituting back, $\underset{x\to 0^{+}}{lim}\frac{e^{-1/x}}{x}=0$, which solves the original problem.

Key concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus specifically designed for evaluating limits that result in indeterminate forms. These forms often appear as $\frac{0}{0}$ or \infty/\infty\.
When a limit falls into these categories, direct evaluation does not provide a clear answer. Here is where L'Hôpital's Rule steps in to help. It simplifies the limit calculation by allowing you to differentiate the numerator and the denominator separately. Once this is done, you can try to recalculate the limit of the resulting fraction.
The rule is particularly handy in simplifying complex limits, making the calculations more manageable and providing an avenue to solutions that might initially seem difficult.
It's important to remember though, that L'Hôpital's Rule isn't always applicable - primarily it's used when you have a clear $\frac{0}{0}$ or \infty/\infty\ form. Knowing when to use it and properly identifying these forms is crucial for success!

• Check if limit leads to an indeterminate form before applying the rule.
• Differentiation must be applicable to both numerator and denominator.
• Keep applying the rule until the limit resolves itself or becomes determinate.

Substitution in Limits

Substitution is a strategic method in limits, especially when dealing with transformations that make complex problems simpler. In essence, substitution is about changing variables to redefine the problem in a more accessible way.
For example, in our original problem, as x approaches 0 from the positive side, the substitution $z=\frac{1}{x}$ transforms the problem to $z\to \mathrm{\infty }$\. This change of variable helps in simplifying expressions that might initially seem daunting or awkward.
Substitution is particularly useful:

• When facing an indeterminate form that needs reformatting to apply rules like L'Hôpital's Rule.
• For simplifying problems in integrals and differential equations, not just limits.
• In resolving function forms that are otherwise difficult to evaluate directly.

However, it requires an eye for identifying which transformations or variables will ease the problem without altering its core values.

Indeterminate Forms

Indeterminate forms are situations in limit problems where substitution and direct calculation yield nonsensical results. They occur when expressions hit forms like $\frac{0}{0}$, \infty - \infty\, or \infty/\infty\.
These forms suggest that further algebraic manipulation or application of calculus concepts is necessary to find the true limit. L'Hôpital's Rule, for instance, is one solution path for indeterminate forms involving division.
Indeterminate forms signal that although the components of the limit may suggest a certain path, the results require a deeper, more technical approach to unravel.
Common types of indeterminate forms include:

• $\frac{0}{0}$
• \infty/\infty\
• \infty - \infty\
• \ 0 \cdot \infty\
• \ 1^\infty, \, 0^0, \, \infty^0\

Recognizing these forms quickly allows students and mathematicians alike to know when to switch techniques and employ more advanced calculus methods.