The image shows three limits that need to be evaluated as $x \to \infty$. Let's break them down:

First Limit:

$\lim_{x \to \infty} \frac{a x^{89} - 7x + 59}{c x^{89} - 59}$

• The highest degree of $x$ in both the numerator and denominator is $x^{89}$.
• As $x \to \infty$, the lower degree terms (like $-7x$, $+59$, and $-59$) become insignificant compared to the highest powers.

So, the limit simplifies to: $\lim_{x \to \infty} \frac{a x^{89}}{c x^{89}} = \frac{a}{c}$

Second Limit:

$\lim_{x \to \infty} \frac{a x^{89} - 7x + 59}{c x^{86} - 59}$

• Here, the highest degree term in the numerator is $x^{89}$, and the highest degree term in the denominator is $x^{86}$.
• When the numerator's degree is higher than the denominator's, the limit tends to infinity.

Thus: $\lim_{x \to \infty} \frac{a x^{89}}{c x^{86}} \sim x^{3} \to \infty$ This limit tends to infinity.

Third Limit:

$\lim_{x \to \infty} \frac{a x^{86} - 7x + 59}{c x^{89} - 59}$

• In this case, the highest degree in the denominator is $x^{89}$, while the highest degree in the numerator is $x^{86}$.
• When the denominator grows faster than the numerator, the entire fraction approaches 0 as $x \to \infty$.

Thus: $\lim_{x \to \infty} \frac{a x^{86}}{c x^{89}} \sim \frac{1}{x^3} \to 0$

Summary of Limits:

1. $\frac{a}{c}$
2. $\infty$
3. $0$

Would you like any more details on these calculations or further clarification?

Here are 5 related questions:

1. How do we determine the dominant term when evaluating limits involving polynomials?
2. What happens if the degree of the numerator is smaller than the denominator in limits to infinity?
3. Why do constants become negligible as $x \to \infty$?
4. Can the limit of a rational function tend to a negative value? Under what circumstances?
5. How do you handle limits with non-polynomial terms like exponential or trigonometric functions?

Tip: Always compare the highest degree terms in both the numerator and denominator when evaluating limits to infinity for rational functions.