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Question:
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If the function
$$
f(x)=\frac{x(x-2)}{x^{2}-4}, x \neq \pm 2
$$
is continuous at $x=2$, then what is $f(2)$ equal to?
Options:
$$
f(x)=\frac{x(x-2)}{x^{2}-4}, x \neq \pm 2
$$
is continuous at $x=2$, then what is $f(2)$ equal to?
Solution:
2088 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{2}$
Let $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}(\mathrm{x}-2)}{\mathrm{x}^{2}-4}=\frac{\mathrm{x}(\mathrm{x}-2)}{(\mathrm{x}-2)(\mathrm{x}+2)}=\frac{x}{x+2}$
Since $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$
$\therefore \quad \lim _{x \rightarrow 2} f(x)=f(2)$
$\Rightarrow \quad \lim _{x \rightarrow 2} \frac{x}{x+2}=f(2)$
$\Rightarrow \quad \mathrm{f}(2)=\frac{2}{4}=\frac{1}{2}$
Since $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$
$\therefore \quad \lim _{x \rightarrow 2} f(x)=f(2)$
$\Rightarrow \quad \lim _{x \rightarrow 2} \frac{x}{x+2}=f(2)$
$\Rightarrow \quad \mathrm{f}(2)=\frac{2}{4}=\frac{1}{2}$
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