91Ó°ÊÓ

To explain why the form \(1^{\infty}\) is indeterminate, and cannot be evaluated by substitution, let's use the example of the limit of \(f(x) = (1+\frac{1}{x})^x\). We will show that as \(x\) approaches infinity, the expression \(f(x)\) takes the form \(1^{\infty}\), but the limit of \(f(x)\) is not \(1\).

We can see that the base of \(f(x)\) is \((1+\frac{1}{x})\), which approaches 1 when \(x\) approaches infinity: \(\lim_{x \to \infty} (1+\frac{1}{x}) = 1\). The exponent of \(f(x)\) is \(x\), which approaches infinity: \(\lim_{x \to \infty}x = \infty\). Therefore, as \(x\) approaches infinity, the expression \(f(x)\) takes the form \(1^{\infty}\).

We will use the limit definition of the exponential function \(e^x\) to calculate the limit of \(f(x)\) as \(x\) approaches infinity: \(\lim_{x \to \infty} (1+\frac{1}{x})^x = e^{\lim_{x \to \infty} x\ln(1+\frac{1}{x})}\) Now, let's consider the new limit inside the exponent: \(\lim_{x \to \infty} x\ln(1+\frac{1}{x})\) We can use the substitution \(u = \frac{1}{x}\), so the limit becomes: \(\lim_{u \to 0} \frac{\ln(1+u)}{u}\) Applying L'Hôpital's rule to this limit: \(\lim_{u \to 0} \frac{\frac{1}{1+u}}{1} = 1\) Now, going back to the original limit: \(\lim_{x \to \infty} (1+\frac{1}{x})^x = e^1 = e\)

From the above steps, we demonstrated that the expression \(f(x)\) takes the form \(1^{\infty}\) when \(x\) approaches infinity, but the actual limit of \(f(x)\) is not equal to 1, but rather equal to \(e\). This shows that we cannot determine the value of an expression of the form \(1^{\infty}\) by mere substitution, as the competing functions (the base approaching 1 and the exponent approaching infinity) could lead to different results depending on the specific functions involved. In this case, the competing functions are: 1. The base approaches 1: \(\lim_{x \to \infty} (1+\frac{1}{x}) = 1\) 2. The exponent approaches infinity: \(\lim_{x \to \infty}x = \infty\) As we have shown, these competing behaviors of the functions in the expression \(f(x)\) lead to an indeterminate form, and we need to use analytical methods such as limits and L'Hôpital's rule to determine the actual value of the expression in the limit.