### Related Formula For a finite limit lim_xto 0 fracf(x)x^n = L$\lim_{x\to 0} \frac{f(x)}{x^n} = L$, the lowest powers of x$x$ below degree n$n$ in the polynomial f(x)$f(x)$ must vanish. ### Core Logic Let the 4th-degree polynomial be: f(x) = ax^4 + bx^3 + cx^2 + dx + e$f(x) = ax^4 + bx^3 + cx^2 + dx + e$ Given: lim_xrightarrow 0 fracax^4 + bx^3 + cx^2 + dx + ex^2 = 5$\lim_{x\rightarrow 0} \frac{ax^4 + bx^3 + cx^2 + dx + e}{x^2} = 5$ For the limit to exist and equal 5, the terms dx$dx$ and e$e$ must be 0$0$, and the coefficient of x^2$x^2$ must be equal to 5: c = 5, quad d = 0, quad e = 0$c = 5, \quad d = 0, \quad e = 0$ Thus, the polynomial simplifies to: f(x) = ax^4 + bx^3 + 5x^2$f(x) = ax^4 + bx^3 + 5x^2$ ### Step 1: Use Extrema Conditions Differentiating f(x)$f(x)$ with respect to x$x$: f'(x) = 4ax^3 + 3bx^2 + 10x = x(4ax^2 + 3bx + 10)$f'(x) = 4ax^3 + 3bx^2 + 10x = x(4ax^2 + 3bx + 10)$ Since f(x)$f(x)$ has extreme values at x=4$x=4$ and x=5$x=5$, f'(4) = 0$f'(4) = 0$ and f'(5) = 0$f'(5) = 0$. This means 4$4$ and 5$5$ are roots of the quadratic factor 4ax^2 + 3bx + 10 = 0$4ax^2 + 3bx + 10 = 0$. ### Step 2: Solve Coefficients Using properties of roots for 4ax^2 + 3bx + 10 = 0$4ax^2 + 3bx + 10 = 0$: textProduct of roots = 4 cdot 5 = 20 = frac104a implies 4a = frac1020 = frac12 implies a = frac18$\text{Product of roots} = 4 \cdot 5 = 20 = \frac{10}{4a} \implies 4a = \frac{10}{20} = \frac{1}{2} \implies a = \frac{1}{8}$ textSum of roots = 4 + 5 = 9 = -frac3b4a$\text{Sum of roots} = 4 + 5 = 9 = -\frac{3b}{4a}$ Substituting 4a = frac12$4a = \frac{1}{2}$: 9 = -frac3b1/2 = -6b implies b = -frac96 = -frac32$9 = -\frac{3b}{1/2} = -6b \implies b = -\frac{9}{6} = -\frac{3}{2}$ Our full polynomial is: f(x) = frac18x^4 - frac32x^3 + 5x^2$f(x) = \frac{1}{8}x^4 - \frac{3}{2}x^3 + 5x^2$ ### Step 3: Calculate f(2) Evaluate at x = 2$x = 2$: f(2) = frac18(2^4) - frac32(2^3) + 5(2^2)$f(2) = \frac{1}{8}(2^4) - \frac{3}{2}(2^3) + 5(2^2)$ = frac168 - frac242 + 20 = 2 - 12 + 20 = 10$= \frac{16}{8} - \frac{24}{2} + 20 = 2 - 12 + 20 = 10$ ### Pattern Recognition Whenever a limit explicitly matches a denominator power x^n$x^n$, it directly yields both the lower-order coefficients as zeroes and the x^n$x^n$ coefficient as the limit value. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Limits, Continuity and Differentiability