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Answer :
To determine the limit of the given function as [tex]\( x \)[/tex] approaches infinity, we analyze the behavior of the polynomial terms in both the numerator and the denominator. The expression is:
[tex]\[
\lim_{x \rightarrow \infty} \frac{12x^2 - 54x^5 - 10 + 45x^3}{7x^5 + 56x^3 + 2x^2 + 16}
\][/tex]
Step 1: Identify the leading terms.
As [tex]\( x \)[/tex] approaches infinity, the leading terms (the terms with the highest power of [tex]\( x \)[/tex]) will dominate the behavior of the polynomial.
- In the numerator, the leading term is [tex]\( -54x^5 \)[/tex].
- In the denominator, the leading term is [tex]\( 7x^5 \)[/tex].
Step 2: Simplify the expression by focusing on the leading terms.
Since both the leading terms of the numerator and the denominator are of the same degree (both are [tex]\( x^5 \)[/tex]), we can approximate the limit by dividing the leading coefficients of these terms.
[tex]\[
\lim_{x \rightarrow \infty} \frac{-54x^5}{7x^5}
\][/tex]
Step 3: Simplify the fraction.
Divide the coefficients:
[tex]\[
\frac{-54}{7}
\][/tex]
Thus, the limit is:
[tex]\[
\boxed{-\frac{54}{7}}
\][/tex]
This result shows that as [tex]\( x \)[/tex] approaches infinity, the function approaches [tex]\( -\frac{54}{7} \)[/tex]. Therefore, the limit is [tex]\( -\frac{54}{7} \)[/tex], a finite number.
[tex]\[
\lim_{x \rightarrow \infty} \frac{12x^2 - 54x^5 - 10 + 45x^3}{7x^5 + 56x^3 + 2x^2 + 16}
\][/tex]
Step 1: Identify the leading terms.
As [tex]\( x \)[/tex] approaches infinity, the leading terms (the terms with the highest power of [tex]\( x \)[/tex]) will dominate the behavior of the polynomial.
- In the numerator, the leading term is [tex]\( -54x^5 \)[/tex].
- In the denominator, the leading term is [tex]\( 7x^5 \)[/tex].
Step 2: Simplify the expression by focusing on the leading terms.
Since both the leading terms of the numerator and the denominator are of the same degree (both are [tex]\( x^5 \)[/tex]), we can approximate the limit by dividing the leading coefficients of these terms.
[tex]\[
\lim_{x \rightarrow \infty} \frac{-54x^5}{7x^5}
\][/tex]
Step 3: Simplify the fraction.
Divide the coefficients:
[tex]\[
\frac{-54}{7}
\][/tex]
Thus, the limit is:
[tex]\[
\boxed{-\frac{54}{7}}
\][/tex]
This result shows that as [tex]\( x \)[/tex] approaches infinity, the function approaches [tex]\( -\frac{54}{7} \)[/tex]. Therefore, the limit is [tex]\( -\frac{54}{7} \)[/tex], a finite number.
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