Check whether the function f ( x ) = 3 x ? 5 is continuous at x = 0 , x = ? 5 and at x = 3 .

Question

Check whether the function $f (x) = 3 x - 5$ is continuous at $x = 0$ , $x = - 5$ and at $x = 3$ .

Smart Achievers · Accepted Answer

(1)

Ans. $f (x)$ is continuous at $x = 3$

Explanation:

$f (x) = 3 x - 5$

At $x = 0$ ,

$f (0) = 3 \times 0 - 5 = - 5$

$\lim_{x \to 0^{+}} f (x) = \lim_{x \to 0} (3 x - 5) = 3 \times 0 - 5$

$= - 5$

$∴$ $\lim_{x \to 0^{+}} f (x) = f (0)$

Therefore, $f$ is continuous at $x = 0$ .

At $x = - 5$ ,

$f (- 5) = 3 \times (- 5) - 5 = - 20$

$\lim_{x \to 5^{+}} f (x) = \lim_{x \to 5} (3 x - 5) = 3 \times (- 5) - 5$

$= - 20$

$∴$ $\lim_{x \to - 5^{+}} f (x) = f (- 5)$

Therefore, $f$ is continuous at $x = - 5$ .

At $x = 3$ , $f (3) = 3 \times (3) - 5 = 4$

$\lim_{x \to 3^{+}} f (x) = \lim_{x \to 3} (3 x - 5) = 3 \times (3) - 5 = 4$

$∴$ $\lim_{x \to 3^{+}} f (x) = f (3)$

Therefore, $f$ is continuous at $x = 3$ .

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