The expression \(frac{infty}{infty}\) is an indeterminate form in mathematics, arising in the context of limits in calculus. The term “infinity” here does not represent a number but rather a concept of unbounded growth. When the numerator and the denominator of a fraction both tend toward infinity, the resulting expression \(frac{infty}{infty}\) lacks a predetermined value. Further analysis is necessary to determine the actual limit.
What Makes a Form Indeterminate
An indeterminate form fails to provide sufficient information to determine the value of a limit solely by substituting the limiting value. This ambiguity stems from a conflict between simultaneous operations. For \(frac{infty}{infty}\), the numerator attempts to make the fraction infinitely large, while the denominator attempts to make it infinitely small.
The outcome depends entirely on the specific functions involved and their relative rates of growth. The limit of a ratio approaching \(frac{infty}{infty}\) could result in zero, a finite non-zero number, infinity, or a value that does not exist. The result cannot be determined without manipulating the expression. The other six common indeterminate forms encountered in calculus are:
- \(frac{0}{0}\)
- \(0 cdot infty\)
- \(infty – infty\)
- \(1^{infty}\)
- \(0^{0}\)
- \(infty^{0}\)
Applying L’Hôpital’s Rule
L’Hôpital’s Rule provides a formal method for resolving indeterminate forms like \(frac{infty}{infty}\) and \(frac{0}{0}\). The rule states that if the limit of a quotient \(f(x)/g(x)\) results in an indeterminate form, the limit of the original ratio equals the limit of the ratio of their derivatives, \(f'(x)/g'(x)\). This technique simplifies complex limit problems by comparing the instantaneous rates of change of the numerator and the denominator.
The rule is applied by differentiating the numerator and the denominator separately. Consider finding the limit of \(frac{x^2}{e^x}\) as \(x\) approaches infinity, which yields \(frac{infty}{infty}\). Applying the rule once gives \(lim_{xtoinfty} frac{2x}{e^x}\), which is still indeterminate. Applying the rule again yields \(lim_{xtoinfty} frac{2}{e^x}\). Since the numerator is constant and the denominator grows without bound, the limit is zero.
Comparing Different Rates of Growth
The ultimate value of a limit yielding \(frac{infty}{infty}\) is determined by which function approaches infinity faster. Analyzing these relative growth rates provides a conceptual understanding that complements L’Hôpital’s Rule. Functions are categorized by their growth hierarchy: exponential functions (\(e^x\)) grow faster than polynomial functions, which grow faster than logarithmic functions (\(ln(x)\)).
For limits involving rational functions (ratios of polynomials), growth rates can be compared using algebraic manipulation. This involves dividing the numerator and denominator by the highest power of the variable present. If the highest power is in the denominator, the limit is zero. If the highest power is in the numerator, the limit is infinity. If the highest powers are the same, the limit is the ratio of their leading coefficients.