Question

Homework. Show that f(x)=5x³+4x²+3 /x³+2x+1 ∈ Θ(x²) using the basic definition (Ad-hoc Calculations) for ∈ R+.

Answer 1

`\[ f(x) = (5x^3 + 4x^2 + 3)/(x^3 + 2x + 1) \in \Theta(x^2) \]`

Step-by-step explanation:

The given expression
`\( f(x) = (5x^3 + 4x^2 + 3)/(x^3 + 2x + 1) \)`can be simplified to
`\( f(x) = (5 + (4)/(x) + (3)/(x^3))/(1 + (2)/(x^2) + (1)/(x^3)) \).` When \( x \) approaches infinity, the terms with the highest powers in the numerator and denominator dominate the expression. In this case, the leading terms are
`\( (4)/(x) \) and \( (2)/(x^2) \)`. Therefore, as \( x \) tends to infinity, the function
`\( f(x) \)` behaves like
`\( (4)/(x) / (2)/(x^2) \)`, which simplifies to
`\( 2x \).` This implies that
`\( f(x) \)`is in the order of
`\( x^2 \)`, leading to the conclusion that
`\( f(x) \in \Theta(x^2) \).`

In more detail, the Big Theta notation
`\( f(x) \in \Theta(g(x)) \)` indicates that there exist positive constants
`\( c_1, c_2, \) and \( k \) such that \( 0 < c_1 \cdot g(x) \leq f(x) \leq c_2 \cdot g(x) \) for all \( x > k \)`. In this case, as
`\( x \)` approaches infinity, the ratio
`\( (f(x))/(x^2) \)` converges to a constant value, satisfying the conditions for
`\( f(x) \in \Theta(x^2) \)`. Therefore, the given function
`\( f(x) \)` grows at the same rate as
`\( x^2 \`), demonstrating the relationship between
`\( f(x) \) and \( x^2 \)` as required.