Question

86. This exercise explores the difference betweenlim(x→∞)(1 + 1/x²)^xandlim(x→∞)(1 + 1/x)^x = ec. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

Accepted Answer

First, identify the function whose limit we want to find: $$f(x) = \left(1 + \frac{1}{x$$^{2}$$}\$$right)^{x}$$. We want to evaluate $$\lim_{x 	o \infty} f(x)$$. Rewrite the expression using the natural logarithm to handle the exponent more easily: consider $$\ln(f(x)) = x \cdot \ln\left(1 + \frac{1}{$$x^{2}$$}ight)$$. Set y$$ = \ln(f(x)) = x \cdot \ln\left(1 + \frac{1}{$$x^{2}$$}ight)$$. To find $$\lim_{x 	o \infty} f(x)$$, we first find $$\lim_{x 	o \infty} y$$ and then exponentiate the result. Rewrite y$ as a quotient to apply l’Hôpital’s Rule: y$$ = \frac{\ln\left(1 + \frac{1}{$$x^{2}$$}ight)}{\frac{1}{x}}$$. As x$$ 	o \infty$$, both numerator and denominator approach 0, so l’Hôpital’s Rule applies. Differentiate numerator and denominator with respect to x$: 
- Numerator derivative: $$\frac{d}{dx} \ln\left(1 + \frac{1}{$$x^{2}$$}ight) = \frac{1}{1 + \frac{1}{$$x^{2}$$}} \cdot \left(-\frac{2}{$$x^{3}$$}ight)$$,
- Denominator derivative: $$\frac{d}{dx} \left(\frac{1}{x}ight) = -\frac{1}{$$x^{2}$$}$$.
Then, compute the limit of the ratio of these derivatives as x$$ 	o \infty$$ to find $$\lim_{x 	o \infty} y$$.

86. This exercise explores the difference between
lim(x→∞)(1 + 1/x²)^x
and
lim(x→∞)(1 + 1/x)^x = e
c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

Verified video answer for a similar problem:

Key Concepts

Limits at Infinity

The Number e and Exponential Limits

l’Hôpital’s Rule