Question

Find the limit of

[ \lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}-4} ]

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Answer

To find the limit

$$ \lim _{x \rightarrow 2} \frac{3x^2 - x - 10}{x^2 - 4} $$

we start by evaluating the expression at $ x = 2 $. This gives us a form $\frac{0}{0}$, which is indeterminate. To handle this, we will simplify the expression.

First, factorize both the numerator and the denominator:

  • The denominator $ x^2 - 4 $ can be factored as $(x - 2)(x + 2)$.
  • The numerator $ 3x^2 - x - 10 $ can be factored as $(x - 2)(3x + 5)$.

Therefore, the expression becomes:

$$ \frac{3x^2 - x - 10}{x^2 - 4} = \frac{(x - 2)(3x + 5)}{(x - 2)(x + 2)} $$

Since we have the common factor $(x - 2)$ in both the numerator and the denominator, we can cancel it out:

$$ \lim_{x \rightarrow 2} \frac{(x - 2)(3x + 5)}{(x - 2)(x + 2)} = \lim_{x \rightarrow 2} \frac{3x + 5}{x + 2} $$

Now, substitute $ x = 2 $ in the simplified expression:

$$ \lim_{x \rightarrow 2} \frac{3x + 5}{x + 2} = \frac{3(2) + 5}{2 + 2} $$

This simplifies to:

$$ \frac{6 + 5}{4} = \frac{11}{4} $$

Thus, the limit is $\frac{11}{4}$.


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