Math  /  Calculus

QuestionQuestion 7 (1 point) limn3n35nn32n2+1\lim _{n \rightarrow \infty} \frac{3 n^{3}-5 n}{n^{3}-2 n^{2}+1} A. -5 B. -2 C. 1 D. 3

Studdy Solution

STEP 1

1. The limit involves a rational function where the degree of the polynomial in the numerator and the denominator are the same.
2. As n n \to \infty , the highest degree terms in the numerator and denominator will dominate the behavior of the function.

STEP 2

1. Identify the highest degree terms in the numerator and the denominator.
2. Simplify the expression by focusing on the highest degree terms.
3. Calculate the limit as n n \to \infty .

STEP 3

Identify the highest degree terms in both the numerator and the denominator:
The numerator is 3n35n 3n^3 - 5n . The highest degree term is 3n3 3n^3 .
The denominator is n32n2+1 n^3 - 2n^2 + 1 . The highest degree term is n3 n^3 .

STEP 4

Simplify the expression by dividing both the numerator and the denominator by n3 n^3 , the highest degree term:
3n35nn32n2+1=3n3/n35n/n3n3/n32n2/n3+1/n3\frac{3n^3 - 5n}{n^3 - 2n^2 + 1} = \frac{3n^3/n^3 - 5n/n^3}{n^3/n^3 - 2n^2/n^3 + 1/n^3}
This simplifies to:
35n212n+1n3\frac{3 - \frac{5}{n^2}}{1 - \frac{2}{n} + \frac{1}{n^3}}

STEP 5

Calculate the limit as n n \to \infty . As n n \to \infty , the terms 5n2\frac{5}{n^2}, 2n\frac{2}{n}, and 1n3\frac{1}{n^3} all approach 0:
limn35n212n+1n3=3010+0=31=3\lim_{n \to \infty} \frac{3 - \frac{5}{n^2}}{1 - \frac{2}{n} + \frac{1}{n^3}} = \frac{3 - 0}{1 - 0 + 0} = \frac{3}{1} = 3
The limit is:
3 \boxed{3}

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