Evaluate the limit
$\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)}$
Answer (1)
Expand the numerator and denominator: ( 2 − x ) ( 9 + 3 x ) = 18 − 3 x − 3 x 2 and ( 3 − 3 x ) ( 4 + 6 x ) = 12 + 6 x − 18 x 2 .
Divide both numerator and denominator by x 2 : 12/ x 2 + 6/ x − 18 18/ x 2 − 3/ x − 3 .
As x → ∞ , terms with x in the denominator approach 0: 0 + 0 − 18 0 − 0 − 3 .
Simplify to find the limit: 6 1 .
Explanation
Problem Setup and Expansion We are asked to evaluate the limit of a rational function as x approaches infinity:
x → ∞ lim ( 3 − 3 x ) ( 4 + 6 x ) ( 2 − x ) ( 9 + 3 x )
To solve this, we will first expand the numerator and the denominator.
Expanding the Numerator Expanding the numerator:
( 2 − x ) ( 9 + 3 x ) = 18 + 6 x − 9 x − 3 x 2 = 18 − 3 x − 3 x 2
Expanding the Denominator Expanding the denominator:
( 3 − 3 x ) ( 4 + 6 x ) = 12 + 18 x − 12 x − 18 x 2 = 12 + 6 x − 18 x 2
Dividing by Highest Power and Evaluating Limit So the limit becomes:
x → ∞ lim 12 + 6 x − 18 x 2 18 − 3 x − 3 x 2
To evaluate the limit as x approaches infinity, we divide both the numerator and the denominator by the highest power of x , which is x 2 :
x → ∞ lim x 2 12 + x 2 6 x − x 2 18 x 2 x 2 18 − x 2 3 x − x 2 3 x 2 = x → ∞ lim x 2 12 + x 6 − 18 x 2 18 − x 3 − 3
As x approaches infinity, the terms x 2 18 , x 3 , x 2 12 , and x 6 all approach 0. Therefore, the limit becomes:
0 + 0 − 18 0 − 0 − 3 = − 18 − 3 = 6 1
Final Answer Therefore, the limit is 6 1 .
Examples
In electrical engineering, when analyzing circuits with complex impedances, you might encounter rational functions whose behavior as frequency approaches infinity needs to be determined. Evaluating such limits helps in understanding the circuit's stability and response at high frequencies, crucial for designing filters and amplifiers.
