49. The value of:
limn→∞ (3n + 5n + 4)/(4 + 2n²)
is _______.
(A) 0
(B) 0.75
(C) 1.5
(D) 3
Evaluate the Limit of (3n + 5n + 4)/(4 + 2n²) as n Tends to Infinity
Understanding the Given Limit Problem
This question is based on the evaluation of a limit of a rational expression as n tends to infinity. The given expression contains polynomial terms in both the numerator and denominator. To determine the limiting value, we need to compare the highest powers of n appearing in the numerator and denominator.
The given limit is:
limn→∞ (3n + 5n + 4)/(4 + 2n²)
The numerator contains terms of degree 1 in n, while the denominator contains a term of degree 2. Since the denominator grows faster than the numerator as n becomes very large, the value of the fraction is expected to approach zero.
Simplifying the Numerator
The numerator of the given expression is:
3n + 5n + 4
The terms 3n and 5n are like terms, so they can be added:
3n + 5n = 8n
Therefore, the numerator becomes:
8n + 4
Hence, the given limit can be rewritten as:
limn→∞ (8n + 4)/(2n² + 4)
Comparing the Highest Powers of n
The highest power of n in the numerator is:
n¹
The highest power of n in the denominator is:
n²
Therefore, the degree of the numerator is 1, while the degree of the denominator is 2.
For a rational expression, if the degree of the numerator is smaller than the degree of the denominator, then the fraction approaches zero as the variable tends to infinity.
Since:
1 < 2
we can conclude that:
limn→∞ (8n + 4)/(2n² + 4) = 0
Detailed Solution by Dividing by the Highest Power of n
To evaluate the limit formally, we divide every term in the numerator and denominator by n², which is the highest power of n present in the entire expression.
Starting with:
limn→∞ (8n + 4)/(2n² + 4)
Dividing the numerator and denominator by n² gives:
limn→∞ [(8n/n²) + (4/n²)] / [(2n²/n²) + (4/n²)]
Simplifying each term:
limn→∞ [(8/n) + (4/n²)] / [2 + (4/n²)]
Now, as n tends to infinity:
8/n → 0
and:
4/n² → 0
Therefore, the limit becomes:
(0 + 0)/(2 + 0)
Thus:
0/2 = 0
Hence, the value of the given limit is:
0
Why the Denominator Dominates the Numerator
The behaviour of the expression can also be understood by examining the dominant terms. For very large values of n, the constant terms become insignificant compared with the terms containing powers of n.
The numerator:
8n + 4
behaves approximately like:
8n
Similarly, the denominator:
2n² + 4
behaves approximately like:
2n²
Therefore, for very large values of n, the fraction behaves like:
8n/2n²
Simplifying:
8n/2n² = 4/n
As n → ∞:
4/n → 0
Therefore, the original expression also approaches 0.
General Rule for Limits of Rational Functions
For a rational expression of the form:
P(n)/Q(n)
where P(n) and Q(n) are polynomials, the limit as n tends to infinity can often be determined by comparing their degrees.
If the degree of the numerator is less than the degree of the denominator, the limit is 0. If both degrees are equal, the limit is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, the expression generally grows without bound or requires further analysis.
In the present problem:
Degree of numerator = 1
Degree of denominator = 2
Since the numerator has the smaller degree, the required limit is:
0
Analysis of All the Given Options
Option (A): 0
This option is correct. After dividing the numerator and denominator by n², the expression becomes:
[(8/n) + (4/n²)]/[2 + (4/n²)]
As n tends to infinity, all terms containing 1/n or 1/n² approach zero. Therefore, the complete expression approaches 0.
Option (B): 0.75
This option is incorrect. A finite non-zero value such as 0.75 would generally arise when the numerator and denominator have the same highest degree and the ratio of their leading coefficients is 0.75. In this problem, the denominator has a higher degree than the numerator, so the limit is zero.
Option (C): 1.5
This option is incorrect. The value 1.5 does not result from the comparison of the dominant terms. The numerator grows linearly as n, while the denominator grows quadratically as n², causing the ratio to approach zero.
Option (D): 3
This option is incorrect. The expression does not approach 3 because the quadratic term 2n² in the denominator grows much faster than the linear term 8n in the numerator. Therefore, the fraction becomes increasingly small as n increases.
Final Answer
The given limit is:
limn→∞ (3n + 5n + 4)/(4 + 2n²)
After simplifying the numerator:
= limn→∞ (8n + 4)/(2n² + 4)
Since the degree of the numerator is less than the degree of the denominator:
limn→∞ (8n + 4)/(2n² + 4) = 0
Correct Option: (A) 0
