Solving Limits of the Form lim_{x to infty} [x^m / e^x] Using L’Hospital’s Rule
Understanding and solving limits of the form lim_{x to infty} [x^m / e^x] where m is a constant, can be quite challenging. This article demonstrates the process using L’Hospital’s Rule to simplify and resolve such limits. We delve into the steps involved and how to handle intermediate indeterminate forms.
The Problem and Basic Formulation
Let's consider the limit:
L lim_{x to infty} frac{x^m}{e^x}
As x approaches infinity, the expression x^m / e^x takes the form of infty / infty, making it an indeterminate form. To resolve this, we can utilize L’Hospital’s Rule, which involves differentiating the numerator and the denominator with respect to x separately.
Applying L’Hospital’s Rule for the First Time
Applying L’Hospital’s Rule for the first time, we differentiate the numerator and the denominator with respect to x:
L lim_{x to infty} frac{mx^{m - 1}}{e^x}
This expression still retains the indeterminate form infty / infty as x approaches infinity, so we need to apply L’Hospital’s Rule again.
Further Differentiation and Indeterminate Forms
Applying L’Hospital’s Rule again, we differentiate the numerator and the denominator once more:
L lim_{x to infty} frac{m(m - 1)x^{m - 2}}{e^x}
As x approaches infinity, the process of differentiating the numerator and the denominator continues to yield intermediate forms of infty / infty. This cycle repeats until the term in the numerator no longer provides an infinite value.
Final Simplification
Eventually, after repeated applications of L’Hospital’s Rule, the term in the numerator will not produce an infinite value when differentiated. The general form of the limit after several differentiations can be represented as:
L lim_{x to infty} frac{m! / x^m}{e^x}
Since e^x grows much faster than any polynomial term, the limit of the above expression as x approaches infinity is 0. Therefore:
L 0
Conclusion
In summary, for any constant m, the limit lim_{x to infty} [x^m / e^x] is 0. This conclusion is reached by repeatedly applying L’Hospital’s Rule to resolve the intermediate indeterminate form of infty / infty. Understanding this process is crucial for solving similar limits involving exponential and polynomial functions.