1. **State the problem:** We need to evaluate the function $$f(x) = \frac{x^3 - 5x^2 + 7x - 2}{x - 2}$$ at the values $x = 1.9, 1.99, 1.999, 2.001, 2.01, 2.1$. 2. **Understand the function:** This is a rational function where the numerator is a cubic polynomial and the denominator is linear. We must be careful at $x=2$ because the denominator becomes zero, which is undefined. 3. **Simplify the function if possible:** Let's try polynomial division or factorization. Divide numerator by denominator: $$x^3 - 5x^2 + 7x - 2 \div (x - 2)$$ Using synthetic division: - Coefficients: 1 (for $x^3$), -5 (for $x^2$), 7 (for $x$), -2 (constant) - Divide by $x - 2$ means use 2 in synthetic division. Steps: - Bring down 1 - Multiply 1 by 2 = 2, add to -5 = -3 - Multiply -3 by 2 = -6, add to 7 = 1 - Multiply 1 by 2 = 2, add to -2 = 0 (remainder) So quotient is: $$x^2 - 3x + 1$$ Therefore: $$\frac{x^3 - 5x^2 + 7x - 2}{x - 2} = x^2 - 3x + 1$$ for all $x \neq 2$. 4. **Evaluate the simplified function at the given values:** Calculate $f(x) = x^2 - 3x + 1$ for each $x$: - For $x=1.9$: $$1.9^2 - 3(1.9) + 1 = 3.61 - 5.7 + 1 = -1.09$$ - For $x=1.99$: $$1.99^2 - 3(1.99) + 1 = 3.9601 - 5.97 + 1 = -1.0099$$ - For $x=1.999$: $$1.999^2 - 3(1.999) + 1 = 3.996001 - 5.997 + 1 = -1.000999$$ - For $x=2.001$: $$2.001^2 - 3(2.001) + 1 = 4.004001 - 6.003 + 1 = -0.998999$$ - For $x=2.01$: $$2.01^2 - 3(2.01) + 1 = 4.0401 - 6.03 + 1 = -0.9899$$ - For $x=2.1$: $$2.1^2 - 3(2.1) + 1 = 4.41 - 6.3 + 1 = -0.89$$ 5. **Summary table of values:** | $x$ | $f(x)$ | |--------|-----------------| | 1.9 | -1.09 | | 1.99 | -1.0099 | | 1.999 | -1.000999 | | 2.001 | -0.998999 | | 2.01 | -0.9899 | | 2.1 | -0.89 | 6. **Note:** The function is undefined at $x=2$ but the limit exists and equals $f(2) = 2^2 - 3(2) + 1 = 4 - 6 + 1 = -1$. This shows the function behaves smoothly around $x=2$ despite the original expression having a zero denominator there.