If f(x) is continuous at x = 3 where f(x) = b | vedclass

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(C) Given that $f(x)$ is continuous at $x = 3$. For a function to be continuous at a point $x = c$,the left-hand limit,right-hand limit,and the value

Vedclass · Accepted Answer

$a - b = \frac{2}{3}$

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(C) Given that $f(x)$ is continuous at $x = 3$. For a function to be continuous at a point $x = c$,the left-hand limit,right-hand limit,and the value of the function at that point must be equal. $	herefore \lim_{x 	o 3^-} f(x) = \lim_{x 	o 3^+} f(x) = f(3)$. Calculating the left-hand limit: $\lim_{x 	o 3^-} f(x) = \lim_{h 	o 0} f(3 - h) = \lim_{h 	o 0} [a(3 - h) + 1] = 3a + 1$. Calculating the right-hand limit: $\lim_{x 	o 3^+} f(x) = \lim_{h 	o 0} f(3 + h) = \lim_{h 	o 0} [b(3 + h) + 3] = 3b + 3$. Since $f(x)$ is continuous at $x = 3$,we equate the limits: $3a + 1 = 3b + 3$. Rearranging the terms: $3a - 3b = 3 - 1$. $3(a - b) = 2$. $a - b = \frac{2}{3}$.

If $f(x)$ is continuous at $x = 3$ where $f(x) = \begin{cases} ax + 1, & \text{for } x \leq 3 \\ bx + 3, & \text{for } x > 3 \end{cases}$,then

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