First slide

Continuity

Question

$If f (x) = \frac{x^{2} - bx + 25}{x^{2} - 7 x + 10} for x \neq 5$ is continuous at x = 5, then the value of f(5) is

Moderate

A

0
B

5
C

10
D

25

Solution

$\begin{array}{l} f (x) = \frac{x^{2} - bx + 25}{x^{2} - 7 x + 10}, x \neq 5 \\ f (x) is continuous at x = 5 only if \lim_{x \to 5} \frac{x^{2} - bx + 25}{x^{2} - 7 x + 10} is finite . \\ Now, x^{2} - 7 x + 10 \to 0 when x \to 5 . \\ Then we must have x^{2} - bx + 25 \to 0 for which b = 10 . \\ Hence, \lim_{x \to 5} \frac{x^{2} - 10 x + 25}{x^{2} - 7 x + 10} = \lim_{x \to 5} \frac{x - 5}{x - 2} = 0 . \end{array}$

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