\[
f(x)=\left\{\begin{array}{ll}
3 x+a & x \geq 3 \\
x^{2}-b & x< 3
\end{array}\right.
\]
\( . a, b \in R \quad \) ، \( \quad \) f \( f(\sqrt{2})=5 \) موجودة وأن \( \lim _{x \rightarrow 3} f(x) \) وكانت

Answer

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Answer

\(a = 3\)

Steps

Step 1 ：\(f(\sqrt{2}) = (\sqrt{2})^2 - b = 2 - b = 5\)

Step 2 ：\(b = -3\)

Step 3 ：\(f(3) = 3(3) + a = 9 + a \)

Step 4 ：\(\lim _{x \rightarrow 3} (x^2 - b) = 3^2 - (-3) = 12 \)

Step 5 ：\(a = 3\)