4. $$ \lim _{x \rightarrow a} \frac{(x-a)}{\left(x^{3}-a^{3}\right)} $$

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The limit is $$ \frac{1}{3a^2} $$. To evaluate the limit $$ \lim_{x \rightarrow a} \frac{(x-a)}{x^3 - a^3}, $$ we first recognize that as $$ x $$ approaches $$ a $$, both the numerator and the denominator approach 0, resulting in an indeterminate form $$ \frac{0}{0} $$. To resolve this, we can factor the denominator. The expression $$ x^3 - a^3 $$ can be factored using the difference of cubes formula: $$ x^3 - a^3 = (x - a)(x^2 + ax + a^2). $$ Substituting this factorization into our limit gives: $$ \lim_{x \rightarrow a} \frac{(x-a)}{(x-a)(x^2 + ax + a^2)}. $$ We can cancel the $$ (x-a) $$ terms in the numerator and denominator (as long as $$ x \neq a $$): $$ \lim_{x \rightarrow a} \frac{1}{x^2 + ax + a^2}. $$ Now, we can directly substitute $$ x = a $$ into the remaining expression: $$ x^2 + ax + a^2 = a^2 + a \cdot a + a^2 = a^2 + a^2 + a^2 = 3a^2. $$ Thus, the limit simplifies to: $$ \frac{1}{3a^2}. $$ Therefore, the final result is: $$ \lim_{x \rightarrow a} \frac{(x-a)}{x^3 - a^3} = \frac{1}{3a^2}. $$

4. \( \lim _{x \rightarrow a} \frac{(x-a)}{\left(x^{3}-a^{3}\right)} \)

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4. \( \lim _{x \rightarrow a} \frac{(x-a)}{\left(x^{3}-a^{3}\right)} \)

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