The function \( f \) is given by \( f(x)=0.1 x^{4}-0.5 x^{3}-3.3 x^{2}+7.7 x-1.99 \) values of \( b \) does \( \lim _{x \rightarrow b} f(x)=2 \) ?

To find the values of \( b \) such that \( \lim_{x \rightarrow b} f(x) = 2 \), we need to set the function equal to 2 and solve for \( x \): \[ 0.1 x^{4} - 0.5 x^{3} - 3.3 x^{2} + 7.7 x - 1.99 = 2 \] First, we can rearrange the equation: \[ 0.1 x^{4} - 0.5 x^{3} - 3.3 x^{2} + 7.7 x - 3.99 = 0 \] Now, we will solve this polynomial equation for \( x \). Let's denote the polynomial as \( P(x) = 0.1 x^{4} - 0.5 x^{3} - 3.3 x^{2} + 7.7 x - 3.99 \) and find its roots. I will use a tool to solve this polynomial equation. Solve the equation by following steps: - step0: Solve for \(x\): \(0.1x^{4}-0.5x^{3}-3.3x^{2}+7.7x-3.99=0\) - step1: Factor the expression: \(\frac{1}{100}\left(10x^{4}-50x^{3}-330x^{2}+770x-399\right)=0\) - step2: Divide both sides: \(10x^{4}-50x^{3}-330x^{2}+770x-399=0\) - step3: Calculate: \(\begin{align}&x\approx 7.999843\\&x\approx 1.048768\\&x\approx 0.951175\\&x\approx -4.999786\end{align}\) - step4: Rewrite: \(x_{1}\approx -4.999786,x_{2}\approx 0.951175,x_{3}\approx 1.048768,x_{4}\approx 7.999843\) The values of \( b \) for which \( \lim_{x \rightarrow b} f(x) = 2 \) are the roots of the equation we solved. The approximate solutions are: 1. \( b_1 \approx -4.999786 \) 2. \( b_2 \approx 0.951175 \) 3. \( b_3 \approx 1.048768 \) 4. \( b_4 \approx 7.999843 \) Thus, the values of \( b \) are approximately: - \( b \approx -5 \) - \( b \approx 0.95 \) - \( b \approx 1.05 \) - \( b \approx 8 \) These are the points where the limit of the function \( f(x) \) approaches 2.