Answer :
To solve the limit problems, we need to analyze the behavior of the functions as [tex]\( x \)[/tex] approaches infinity. Here’s a step-by-step breakdown:
### First Limit
[tex]\[
\lim_{x \to \infty} \frac{9x^{10} + 6x^4 - 9}{7x^7 - 8x - 6}
\][/tex]
1. Identify the Leading Terms:
- In the numerator, the leading term (the term with the highest power) is [tex]\( 9x^{10} \)[/tex].
- In the denominator, the leading term is [tex]\( 7x^7 \)[/tex].
2. Simplify the Expression:
- Factor out the highest power of [tex]\( x \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{9x^{10} \left(1 + \frac{6}{9x^6} - \frac{9}{9x^{10}}\right)}{7x^7 \left(1 - \frac{8}{7x^6} - \frac{6}{7x^7}\right)} = \frac{9x^{10}}{7x^7} = \frac{9}{7}x^3
\][/tex]
3. Evaluate the Limit:
- As [tex]\( x \to \infty \)[/tex], [tex]\( \frac{9}{7}x^3 \to \infty \)[/tex].
Thus, the first limit approaches infinity.
### Second Limit
[tex]\[
\lim_{x \to \infty} \frac{30x^7 + 9x^4 + 6}{6x^5 - 7x^4 + 9}
\][/tex]
1. Identify the Leading Terms:
- In the numerator, the leading term is [tex]\( 30x^7 \)[/tex].
- In the denominator, the leading term is [tex]\( 6x^5 \)[/tex].
2. Simplify the Expression:
- Factor out the highest power of [tex]\( x \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{30x^7 \left(1 + \frac{9}{30x^3} + \frac{6}{30x^7}\right)}{6x^5 \left(1 - \frac{7}{6x} + \frac{9}{6x^5}\right)} = \frac{30x^7}{6x^5} = 5x^2
\][/tex]
3. Evaluate the Limit:
- As [tex]\( x \to \infty \)[/tex], [tex]\( 5x^2 \to \infty \)[/tex].
Thus, the second limit also approaches infinity.
In conclusion, both limits tend to infinity as [tex]\( x \)[/tex] approaches infinity, indicating that they do not have finite real number results.
### First Limit
[tex]\[
\lim_{x \to \infty} \frac{9x^{10} + 6x^4 - 9}{7x^7 - 8x - 6}
\][/tex]
1. Identify the Leading Terms:
- In the numerator, the leading term (the term with the highest power) is [tex]\( 9x^{10} \)[/tex].
- In the denominator, the leading term is [tex]\( 7x^7 \)[/tex].
2. Simplify the Expression:
- Factor out the highest power of [tex]\( x \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{9x^{10} \left(1 + \frac{6}{9x^6} - \frac{9}{9x^{10}}\right)}{7x^7 \left(1 - \frac{8}{7x^6} - \frac{6}{7x^7}\right)} = \frac{9x^{10}}{7x^7} = \frac{9}{7}x^3
\][/tex]
3. Evaluate the Limit:
- As [tex]\( x \to \infty \)[/tex], [tex]\( \frac{9}{7}x^3 \to \infty \)[/tex].
Thus, the first limit approaches infinity.
### Second Limit
[tex]\[
\lim_{x \to \infty} \frac{30x^7 + 9x^4 + 6}{6x^5 - 7x^4 + 9}
\][/tex]
1. Identify the Leading Terms:
- In the numerator, the leading term is [tex]\( 30x^7 \)[/tex].
- In the denominator, the leading term is [tex]\( 6x^5 \)[/tex].
2. Simplify the Expression:
- Factor out the highest power of [tex]\( x \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{30x^7 \left(1 + \frac{9}{30x^3} + \frac{6}{30x^7}\right)}{6x^5 \left(1 - \frac{7}{6x} + \frac{9}{6x^5}\right)} = \frac{30x^7}{6x^5} = 5x^2
\][/tex]
3. Evaluate the Limit:
- As [tex]\( x \to \infty \)[/tex], [tex]\( 5x^2 \to \infty \)[/tex].
Thus, the second limit also approaches infinity.
In conclusion, both limits tend to infinity as [tex]\( x \)[/tex] approaches infinity, indicating that they do not have finite real number results.
