Question:
Consider
\[f(x)=\left\{\begin{array}{ll} \frac{x^{2}-x-6}{x+2} & x \neq-2 \\ -5 & x=-2 \end{array}\right.\]
Check whether the function \(f(x)\) is continuous at \(x= -2\).
Answer:
$$f(-2)=5$$
$$\lim_{x \to -2}f(x)=\lim_{x \to-2}\frac{x^2-x-6}{x+2}$$
$$=\lim_{x \to -2}f(x)=\lim_{x \to-2}\frac{(x-3)(x+2)}{x+2}$$
$$=\lim_{x \to -2}(x-3)=f(-2)=-5$$
Hence \(f(x)\) is continuous at \(x=-2\)
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