Solving Limits with L’Hospital’s Rule | A Powerful Technique in Calculus

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

The L’Hospital’s rule, also known as L’Hospital’s shortcut, is a powerful technique used in calculus to evaluate limits of indeterminate forms

The L’Hospital’s rule, also known as L’Hospital’s shortcut, is a powerful technique used in calculus to evaluate limits of indeterminate forms. Indeterminate forms occur when you have a fraction or expression that doesn’t have a unique limit as the variables approach a certain value, such as 0/0 or infinity/infinity.

Specifically, L’Hospital’s rule can be applied to evaluate limits of the form f(x)/g(x) as x approaches a certain value (usually 0 or infinity), where f(x) and g(x) are differentiable functions. It states that if the limit of the ratio f(x)/g(x) as x approaches the value in consideration is an indeterminate form, then the limit can be determined by taking the derivative of both the numerator and the denominator separately, and then evaluating the new ratio of derivatives.

To apply L’Hospital’s rule to a polynomial/exponential situation, consider the limit of a polynomial function divided by an exponential function. For example, let’s say we have the limit:

lim(x → 0) (x^2)/(e^x)

Here, x^2 is a polynomial function and e^x is an exponential function. Since both the numerator and the denominator approach zero as x approaches 0, this limit falls under the indeterminate form of 0/0.

To find the value of this limit using L’Hospital’s rule, you differentiate the numerator and the denominator separately. The derivative of x^2 is 2x, and the derivative of e^x is simply e^x. Then, you evaluate the new ratio of derivatives:

lim(x → 0) (2x)/(e^x)

Now, you can directly substitute x = 0 into this expression to find the limit:

lim(x → 0) (2(0))/(e^0) = 0/1 = 0

Hence, using L’Hospital’s rule, we determined that the limit of (x^2)/(e^x) as x approaches 0 is equal to 0.

Similarly, you can apply L’Hospital’s rule to evaluate limits of the form f(x)/g(x) as x approaches infinity, where f(x) and g(x) are differentiable functions. In this case, you would take the derivative of the numerator and denominator and evaluate the limit using the new ratio of derivatives.

Remember, L’Hospital’s rule is applicable when you have indeterminate forms, and it provides a way to simplify the evaluation of limits in such cases. However, it is important to use the rule judiciously and ensure that the conditions for its application are satisfied.

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