problem is: Use continuity to evaluate the limit. $$ \lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{5 x^{2}-10 x}\right) $$

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To evaluate the limit

$$
\lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{5 x^{2}-10 x}\right),
$$

we can start by simplifying the expression inside the arctan function.

1. **Simplify the expression**: The expression inside the limit is

$$
\frac{x^{2}-4}{5 x^{2}-10 x}.
$$

Notice that $$x^2 - 4$$ can be factored as $$(x - 2)(x + 2)$$ and $$5x^2 - 10x$$ can be factored as $$5x(x - 2)$$. Thus, we can rewrite the expression as:

$$
\frac{(x - 2)(x + 2)}{5x(x - 2)}.
$$

2. **Cancel common factors**: Since $$x \rightarrow 2$$ and $$x - 2$$ is a common factor, we can cancel it (as long as $$x \neq 2$$):

$$
\frac{x + 2}{5x}.
$$

3. **Evaluate the limit**: Now we can substitute $$x = 2$$ into the simplified expression:

$$
\lim_{x \rightarrow 2} \frac{x + 2}{5x} = \frac{2 + 2}{5 \cdot 2} = \frac{4}{10} = \frac{2}{5}.
$$

4. **Apply continuity of arctan**: Since the arctan function is continuous, we can now evaluate the limit of the arctan function:

$$
\lim_{x \rightarrow 2} \arctan \left(\frac{x + 2}{5x}\right) = \arctan\left(\frac{2}{5}\right).
$$

Thus, the final result is:

$$
\lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{5 x^{2}-10 x}\right) = \arctan\left(\frac{2}{5}\right).
$$
The limit is $$ \arctan\left(\frac{2}{5}\right) $$.

problem is: Use continuity to evaluate the limit. \( \lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{5 x^{2}-10 x}\right) \)

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