Test whether the function is continuous or discontinuous at x = 2. Exp | Basic Mathematics 2081.2

Basic Mathematics2081.25 marksNumerical: continuity test at pointContinuity definition

Test whether the function $f(x) = \begin{cases} \frac{x^2 - 4}{x - 2} & \text{if } x \ne 2 \\ 4 & \text{if } x = 2 \end{cases}$ is continuous or discontinuous at $x = 2$ . Explain. [5]

Test whether the function $$f(x) = \begin{cases} \frac{x^2 - 4}{x - 2} & \text{if } x \ne 2 \ 4 & \text{if } x = 2 \end{cases}$$ is continuous or discontinuous at $x = 2$. Explain. [5]

$f(x) = \begin{cases} \dfrac{x^2 - 4}{x - 2} & \text{if } x \ne 2 \\[2mm] 4 & \text{if } x = 2 \end{cases}$ Point to test: $x = 2$ . A function $f$ is continuous at $x = a$ if: 1. $f(a)$ is defined, 2. $\lim{x \to a} f(x)$ exists, 3. $\lim{x \to a} f(x) = f(a)$ . By definition: $f(2) = 4$ So

$$f(x) = \begin{cases} \dfrac{x^2 - 4}{x - 2} & \text{if } x \ne 2 \[2mm] 4 & \text{if } x = 2 \end{cases}$$ Point to test: $x = 2$. A function $f$ is continuous at $x = a$ if: 1. $f(a)$ is defined, 2. $\lim{x \to a} f(x)$ exists, 3. $\lim{x \to a} f(x) = f(a)$. By definition: $$f(2) = 4$$ So

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Test whether the function is continuous or discontinuous at x = 2. Explain.