Expert Answer to "Use the definition of continuity and the properties..."

varaderiyw

Answered question

2021-11-18

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

f (x) = \frac{2 x + 3}{x - 2}, (2, \infty)

Othrom

Beginner2021-11-19Added 15 answers

f(x) is continous at x=a if and only if

lim_{x \to a} f (x) = f (a)

lim_{x \to a} f (x) = lim_{x \to a} \frac{2 x + 3}{x - 2}

Divisionlaw for lomits

lim_{x \to a} \frac{2 x + 3}{x - 2} = \frac{lim_{x \to a} 2 x + 3}{lim_{x \to a} x - 2}

\frac{lim_{x \to a} 2 x + 3}{lim_{x \to a} x - 2} = \frac{lim_{x \to a} 2 x + lim_{x \to a} 3}{lim_{x \to a} x - 2}

\frac{lim_{x \to a} 2 x + lim_{x \to a} 3}{lim_{x \to a} x - lim_{x \to a} 2} = \frac{2 \cdot lim_{x \to a} x + lim_{x \to a} 3}{lim_{x \to a} x - lim_{x \to a} 2}

\frac{2 \cdot lim_{x \to a} x + lim_{x \to a} 3}{lim_{x \to a} x - lim_{x \to a} 2} = \frac{2 a + 3}{a - 2}

\frac{2 a + 3}{a - 2} = f (a)

Hence proved that f(x) is continuous for all

a \neq 2

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