Final answer:
The limit of the function as x approaches infinity is 7/4, which is found by considering the leading terms of the numerator and denominator and simplifying.
Step-by-step explanation:
To evaluate the limit lim (x → ∞) (8x² - x²) / (4x² + 5x - 3), we first note that as x approaches infinity, the highest power terms in the numerator and denominator will dominate the behavior of the function.
Therefore, the x² terms in the numerator (8x² and -x²) and the x² term in the denominator (4x²) are the most significant.
We can simplify the expression by dividing both the numerator and the denominator by x², the highest power of x present in the expression:
(8 - 1)/(4 + 5/x - 3/x²)
As x approaches infinity, the terms 5/x and 3/x² approach zero. What we are left with is the leading coefficients of the x² terms:
(8 - 1) / 4
This simplifies to 7/4. So, the limit of the function as x approaches infinity is 7/4.