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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
Question
Prove that the function f(x) = 5x - 3 is continuous at x = 0, at x = -3 and at x = 5
✓

Answer

The given function is $f(x) = 5x - 3, ~~at~~ x = 0$,
Clearly, $f(0) = 5 \times0 - 3 = -3$
$\mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to 0} (5x - 3) = 5 \times 0 - 3 = - 3$
Thus, $\mathop {\lim }\limits_{x \to 0} f(x) = f(0)$
Therefore, f is continuous at x = 0
At x = -3, f(-3) = $5 \times(-3)-3=-18$
$\mathop {\lim }\limits_{x \to - 3} f(x) = \mathop {\lim }\limits_{x \to - 3} (5x - 3) = 5 \times ( - 3) - 3 = - 18$
Thus, $\mathop {\lim }\limits_{x \to - 3} f(x) = f( - 3)$
Therefore, f is continuous at x = -3
$At~~ x = 5, f(5) =  5\times5-3=22$
$\mathop {\lim }\limits_{x \to 5} f(x) = \mathop {\lim }\limits_{x \to 5} (5x - 3) = 5 \times 5 - 3 = 22$
Thus, $\mathop {\lim }\limits_{x \to 5} f(x) = f(5)$
Therefore, f is continuous at = 5

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