Question

what is the limit of 3x^2+3ax-2a^2 over x^2-a^2 as x approaches to a.

Answer 1

The limit is given by:

$\lim_{x \to a} \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2}$

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$\\$ Left Hand Limit:

$\lim_{x \to a^{-} } \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2}$

From the function of limit and the limit value, we can say that the function decreases without a bound. So, we can write:

$\lim_{x \to a^{-} } \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2} = - \infty$

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Right Hand Limit:

$\lim_{x \to a^{+} } \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2}$

From the function of limit and the limit value, we can say that the function grows without a bound. So, we can write:

$\lim_{x \to a^{+} } \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2} = \infty$

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Since:

$\lim_{x \to a^{-} } \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2} \neq \lim_{x \to a^{+} } \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2}$

Hence:

$\lim_{x \to a} \frac{3x^2 + 3ax - 2a^2}{x^2 - a^2}$ does not exist.