Question

name: date:
specific limits, general limits
part a
consider the following rational function, $g(x)=\frac{3x^2}{x^2 - 4}$.

1. find the following descriptions of $g(x)$:

$\lim_{x\to 2^-}g(x)=$ $\lim_{x\to 2^+}g(x)=$ $\lim_{x\to 2}g(x)=$
$\lim_{x\to -2^-}g(x)=$ $\lim_{x\to -2^+}g(x)=$ $\lim_{x\to -2}g(x)=$
$\lim_{x\to \infty}g(x)=$ $\lim_{x\to -\infty}g(x)=$
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page 3 of 12

Explanation:

Step1: Analyze $\lim_{x \to 2^+} g(x)$

As $x$ approaches 2 from the right, the graph of $g(x)$ rises to $+\infty$.
$\lim_{x \to 2^+} g(x) = +\infty$

Step2: Analyze $\lim_{x \to 2^-} g(x)$

As $x$ approaches 2 from the left, the graph of $g(x)$ falls to $-\infty$.
$\lim_{x \to 2^-} g(x) = -\infty$

Step3: Analyze $\lim_{x \to 2} g(x)$

Since the left-hand and right-hand limits at $x=2$ are not equal, the two-sided limit does not exist.
$\lim_{x \to 2} g(x) = \text{Does Not Exist (DNE)}$

Step4: Analyze $\lim_{x \to -2^+} g(x)$

As $x$ approaches -2 from the right, the graph of $g(x)$ falls to $-\infty$.
$\lim_{x \to -2^+} g(x) = -\infty$

Step5: Analyze $\lim_{x \to -2^-} g(x)$

As $x$ approaches -2 from the left, the graph of $g(x)$ rises to $+\infty$.
$\lim_{x \to -2^-} g(x) = +\infty$

Step6: Analyze $\lim_{x \to -2} g(x)$

Since the left-hand and right-hand limits at $x=-2$ are not equal, the two-sided limit does not exist.
$\lim_{x \to -2} g(x) = \text{Does Not Exist (DNE)}$

Step7: Analyze $\lim_{x \to \infty} g(x)$

As $x$ approaches $+\infty$, the graph approaches the horizontal asymptote $y=3$.
$\lim_{x \to \infty} g(x) = 3$

Step8: Analyze $\lim_{x \to -\infty} g(x)$

As $x$ approaches $-\infty$, the graph approaches the horizontal asymptote $y=3$.
$\lim_{x \to -\infty} g(x) = 3$

Answer:

$\lim_{x \to 2^+} g(x) = +\infty$
$\lim_{x \to 2^-} g(x) = -\infty$
$\lim_{x \to 2} g(x) = \text{Does Not Exist (DNE)}$
$\lim_{x \to -2^+} g(x) = -\infty$
$\lim_{x \to -2^-} g(x) = +\infty$
$\lim_{x \to -2} g(x) = \text{Does Not Exist (DNE)}$
$\lim_{x \to \infty} g(x) = 3$
$\lim_{x \to -\infty} g(x) = 3$