L’hopital’s Rule Calculator
Find the values of the undetermined limits using L’hopital’s Rule Calculator. This tool will differentiate the function and apply the limit to find its value.
How to use this tool?
The steps to use the L’hopital’s Rule calculator are as follows.
- Enter the functions. Add any math keys by clicking on the keyboard icon.
- Select the variable and the side of the limit.
- Lastly, enter the value of the limit.
- Click Calculate to find the answer.
What is the L’hopital’s Rule?
L’hopital’s rule is a technique used in calculus to find the value of undetermined forms like 0/0 or ∞/∞.
An indeterminate form is a mathematical expression that doesn't have a clear and direct value. Common examples include 0/0, ∞/∞, 0 x ∞, ∞ - ∞, 00, and ∞0. An indeterminate form doesn't imply that the limit doesn't exist or that it doesn't have a specific value.
Instead, it means that the limit cannot be directly determined in its current form, and additional steps are required to find its value. For example, a limit that results in the form 0/0 is indeterminate because we cannot directly conclude its value from the form itself.
Formula of the L’hopital’s rule:
Suppose f(x) and g(x) are functions that are differentiable near a point a (except possibly at a) and these three conditions are met:
- lim x→a f(x) = 0 or lim x→a f(x) = ∞
- lim x→a g(x)=0 or lim x→a g(x) = ∞
- lim x→a f’(x)/g’(x) exits or is equal to ∞.
This means that
lim x→a f’(x)/g’(x) = lim x→a f(x)/g(x)
How to use the L’hopital’s rule?
To apply L'Hôpital's rule, follow these steps:
- Verify that the limit is in the form 0/0 or ∞/∞.
- Differentiate the numerator and denominator separately.
- Calculate the limit of the quotient of the derivatives.
- If the limit obtained in step 3 is still indeterminate, apply L'Hôpital's rule again until you find a determinate form.
Example 1:
Find the limit of \[\lim_{x \to \infty} \frac{e^x}{x^2}\].
Direct Substitution:
If we substitute x=∞ into the expression, we get
\[ \lim_{{x \to \infty}} \frac{{e^x}}{{x^2}} = \frac{{e^\infty}}{{\infty^2}} = \frac{\infty}{\infty}. \]
Which is an indeterminate form ∞/∞.
Applying L'Hôpital's Rule:
First, we need to find the derivatives of the numerator and the denominator.
\[ \frac{{d}}{{dx}} e^x = e^x \quad \text{and} \quad \frac{{d}}{{dx}} x^2 = 2x. \]
Now, we evaluate the limit of the quotient of the derivatives:
\[ \lim_{{x \to \infty}} \frac{{e^x}}{{2x}}. \]
On differentiating we find that it is still in an indeterminate form ∞/∞, so apply
L'Hôpital's rule again:
\[ \frac{{d}}{{dx}} e^x = e^x \quad \text{and} \quad \frac{{d}}{{dx}} 2x = 2. \]
Evaluate the limit of the quotient of the new derivatives:
\[ \lim_{{x \to \infty}} \frac{{e^x}}{2} = \infty. \]
Conclusion:
\[ \lim_{{x \to \infty}} \frac{{e^x}}{{x^2}} = \infty. \]
The limit is infinity.
Example 2:
Evaluate the following limit:
\[\lim_{x \to 0} \frac{e^{x^2} - 1}{x^2}\]
Direct Substitution:
If we substitute katx=0 into the expression, we get
\[
\frac{e^{0^2} - 1}{0^2} = \frac{1-1}{0} = \frac{0}{0}
\]
which is an indeterminate form 0/0. So, we can apply L'Hôpital's rule.
Applying L'Hôpital's Rule:
First, we need to find the derivatives of the numerator and the denominator.
\[\frac{d}{dx} (e^{x^2} - 1) = 2x e^{x^2} \quad \text{and} \quad \frac{d}{dx} x^2 = 2x\]
Now, we evaluate the limit of the quotient of the derivatives:
\[\lim_{x \to 0} \frac{2x e^{x^2}}{2x} = \lim_{x \to 0} e^{x^2} = e^{0^2} = e^0 = 1\]
So,
\[\lim_{x \to 0} \frac{e^{x^2} - 1}{x^2} = 1\]
Applications of L’hopital’s rule:
Here are some common applications:
Analysis: Used in real and complex analysis to study the behavior of functions near singular points or asymptotes.
Physics: Calculating instantaneous rates of change in kinematics, such as finding velocity and acceleration from position-time graphs.
Engineering: Analyzing the behavior of circuits and systems in electrical engineering, particularly in the study of signals and systems.
References:
Khan Academy. (n.d.). L’Hoptial’s Rule introduction. Khan Academy.