Simplifying Limits with L’Hôpital’s Rule | A Powerful Technique in Calculus

L’hospital shortcut: exponential/polynomial (infinity/number)

L’Hôpital’s rule, also known as the L’Hôpital shortcut, is a powerful technique in calculus used for evaluating limits involving indeterminate forms, specifically the form of “exponential function divided by a polynomial function” as the input variable approaches either infinity or a finite number

L’Hôpital’s rule, also known as the L’Hôpital shortcut, is a powerful technique in calculus used for evaluating limits involving indeterminate forms, specifically the form of “exponential function divided by a polynomial function” as the input variable approaches either infinity or a finite number.

Let’s consider the general form of the indeterminate limit:
lim(x→a) [f(x)/g(x)].

In the case of the L’Hôpital shortcut for exponential/polynomial limits, we have f(x) as an exponential function and g(x) as a polynomial function.

There are two different scenarios we can encounter: when the limit involves either the input variable approaching infinity (lim(x→∞)) or the input variable approaching a finite number (lim(x→a)).

1. Exponential/polynomial limit as x approaches infinity (lim(x→∞)):
If we have an indeterminate form of the type ∞/∞ or 0/0, we can apply L’Hôpital’s rule to simplify the limit.

The steps to apply L’Hôpital’s rule in this case are as follows:
1. Differentiate both f(x) and g(x) with respect to x.
2. Take the derivative of the numerator and the denominator separately.
3. Calculate the new limit, which is the limit of the derivatives.
4. If the indeterminate form persists, repeat the process until the form resolves to a determinate answer.

For example, let’s say we have the limit:
lim(x→∞) [e^x / (1 + x^2)].

We can use L’Hôpital’s rule in the following way:
1. Differentiate the numerator: d/dx (e^x) = e^x.
2. Differentiate the denominator: d/dx (1 + x^2) = 2x.
3. Calculate the new limit: lim(x→∞) [e^x / 2x].
4. If this limit still results in the indeterminate form, we can repeat the steps until we obtain a determinate answer.

2. Exponential/polynomial limit as x approaches a finite number (lim(x→a)):
In this case, we can adapt L’Hôpital’s rule when we have the indeterminate form 0/0.

The steps to apply L’Hôpital’s rule in this case are slightly different:
1. Evaluate the limits of both the numerator f(x) and denominator g(x) as x approaches a.
2. If the limits of both f(x) and g(x) exist and the denominator limit is nonzero, then we can apply L’Hôpital’s rule.
3. Differentiate both f(x) and g(x) with respect to x.
4. Calculate the new limit, which is the limit of the derivatives.
5. If the indeterminate form persists, repeat the process until the form resolves to a determinate answer.

For example, let’s say we have the limit:
lim(x→2) [(e^x – e^2) / (x – 2)].

We can use L’Hôpital’s rule in the following way:
1. Evaluate the limits at x = 2:
– Numerator: e^x – e^2 = e^2 – e^2 = 0.
– Denominator: x – 2 = 2 – 2 = 0.
2. As both limits exist and the denominator limit is nonzero, we can apply L’Hôpital’s rule.
3. Differentiate the numerator: d/dx (e^x – e^2) = e^x.
4. Differentiate the denominator: d/dx (x – 2) = 1.
5. Calculate the new limit: lim(x→2) (e^x / 1) = e^2.

By applying L’Hôpital’s rule, we can often simplify complex limits involving exponential and polynomial functions as the input variable approaches infinity or a finite number, enabling us to find their determinate values.

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