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A designer studies the ratio R(x) = (|x + 2|)/(x² - 4) for x near -2, where both the top and the…

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A designer studies the ratio $R(x) = \dfrac{|x + 2|}{x^2 - 4}$ for $x$ near $-2$ , where both the top and the bottom vanish. The modulus makes the two sides behave differently. Find the right-hand limit of $R(x)$ as $x$ tends to $-2$ from above. Then find the left-hand limit from below. Give both as decimals. Say whether the limit at $-2$ exists.

Need a reminder? The key ideaShow

Squeeze: if $g \le f \le h$ and $g, h \to L$ , then $f \to L$ . Split $|x - a|$ at $a$ . For the greatest integer, $[x]$ is the largest integer not above $x$ , and $[-3.2] = -4$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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