91Ó°ÊÓ

When we talk about limits at infinity, we're investigating how a function behaves as the input grows larger and larger, either positively or negatively. For expression \( \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}}{x^{2}} \), we are interested in seeing how this function approaches a particular value when \( x \) becomes infinitely large.

In our case, as \( x \rightarrow \infty \), the numerator, \( \sqrt{1+x^2} \), behaves like \( x \) because \( x^2 \) dominates in the square root. The denominator \( x^2 \) is obviously \( x^2 \). This brings the intuitive understanding that \( \frac{\text{top}}{\text{bottom}} \) starts behaving like \( \frac{x}{x^2} = \frac{1}{x} \), which approaches 0. Hence, the limit is 0. This thinking will guide how limits can be manipulated for simplicity and understanding.

Indeterminate forms are expressions that don't initially have a clear value. These forms frequently occur in calculus, especially in limits. Common indeterminate forms include 0/0, ∞/∞, and others like 1^∞ and 0^0.

In our exercise, we face the form \( \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}}{x^{2}} \). As \( x \) grows, both the numerator and the denominator become infinite, leading to the indeterminate form ∞/∞. This suggests that the direct computation of limits is not straightforward here.

To resolve such forms, techniques like L'Hopital's rule can be applied, where permissible. These rules allow us to take derivatives of the numerator and denominator to evaluate limits more effectively. It's like finding a new perspective that simplifies the complexity of the original scenario.

Differentiation techniques play a critical role in calculus, particularly when evaluating limits involving indeterminate forms. L'Hopital's rule is a powerful tool in this context. It gives us a way to handle forms like 0/0 or ∞/∞ by differentiating the top and bottom separately.

In the given problem, the form \( \frac{\infty}{\infty} \) was simplified using differentiation. Differentiating the numerator \( \sqrt{1+x^2} \) gave \( 0.5(1 + x^2)^{-0.5} \times 2x \). The derivative of the denominator \( x^2 \) is \( 2x \).

Applying differentiation allowed us to rewrite the limit as \( \lim _{x \rightarrow \infty} (1+x^{2})^{-0.5}x \). Upon further simplification and analysis, we see the limit approaching 0. Using these techniques can make complex limits more approachable and logical.