Final answer:
The limit as x approaches 3 is 0.6923 and the limit as x approaches infinity is 0.25.
Step-by-step explanation:
The question is asking for the limit of the expression (x² / (4x + 1)) as x approaches 3 and as x approaches infinity. To find the limit as x approaches 3, we substitute x = 3 into the expression:
(3²) / (4(3) + 1) = 9 / 13 = 0.6923
To find the limit as x approaches infinity, we divide each term in the expression by the highest degree of x (which is x²):
(x² / x²) / (4x / x² + 1 / x²)
As x approaches infinity, all terms with a positive power of x become negligible compared to the term with the highest power of x, so we can ignore them. This leaves us with:
1 / (4(1) + 0) = 1 / 4 = 0.25