Functions
Continuity & Limits
MMTS_Full_Test_19
Grade 12
Question:
If $f(x)=x^2+ax+3$ and $g(x)=x+b$, where $F(x)=\lim_{n\to\infty}\dfrac{f(x)+(x^2)^n g(x)}{1+(x^2)^n}$. If $F(x)$ is continuous at $x=1$ and $x=-1$, then find the value of $(a^2+b^2)$
Step-by-Step Solution
Key Concept: At $|x|=1$: $F(x)$ depends on limit; continuity forces specific values
At $x=1$: $F(1)=g(1)=1+b$. Continuity: $f(1)=1+a+3=g(1)=1+b\Rightarrow b=a+3$. At $x=-1$: similarly $a-b=3$... Solving: $a^2+b^2=17$.
Correct Answer: 17