To find the limit $\lim_{{x \to \infty}} \left( 5 - \frac{3}{x^2} \right)$, we will use the properties of limits step-by-step.

1. Separate the Limit: We can separate the limit into two parts using the property of limits that allows us to handle sums and differences separately: $\lim_{{x \to \infty}} \left( 5 - \frac{3}{x^2} \right) = \lim_{{x \to \infty}} 5 - \lim_{{x \to \infty}} \frac{3}{x^2}$

2. Evaluate Each Limit Separately:

   • For the first limit: $\lim_{{x \to \infty}} 5 = 5$ Since 5 is a constant, its limit as $x$ approaches infinity is just 5.

   • For the second limit: $\lim_{{x \to \infty}} \frac{3}{x^2}$ As $x$ approaches infinity, the denominator $x^2$ becomes very large, causing the entire fraction to approach zero. Therefore: $\lim_{{x \to \infty}} \frac{3}{x^2} = 0$

3. Combine the Results: Now, combining the results of both limits: $\lim_{{x \to \infty}} \left( 5 - \frac{3}{x^2} \right) = 5 - 0 = 5$

Hence, the limit is:

$\lim_{{x \to \infty}} \left( 5 - \frac{3}{x^2} \right) = 5$

Would you like more details or have any questions? Here are 5 related questions to expand on this topic:

1. How does the limit of $\frac{1}{x}$ behave as $x$ approaches infinity or zero?
2. What happens to $\lim_{{x \to \infty}} \left( \frac{1}{x^n} \right)$ for different values of $n$?
3. Can limits be used to determine horizontal asymptotes of functions? How?
4. How would the limit change if the term $-\frac{3}{x^2}$ were replaced with $-\frac{3}{x}$?
5. What are the formal rules and properties for evaluating limits involving sums and differences?

Tip: When evaluating limits as $x$ approaches infinity, terms that decrease faster (e.g., $\frac{1}{x^2}$) will tend to zero more rapidly than those that decrease slower (e.g., $\frac{1}{x}$).