Combining limits using sum, product, and quotient rules

First evaluate the numerator's limit: $\lim_{x\to a}[3f(x) - 2g(x)] = 3\lim_{x\to a} f(x) - 2\lim_{x\to a} g(x) = 3(2) - 2(5) = 6 - 10 = -4$, using the constant multiple rule and difference rule.

Next evaluate the denominator's limit using the product rule: $\lim_{x\to a}[f(x)g(x)] = \left(\lim_{x\to a} f(x)\right)\left(\lim_{x\to a} g(x)\right) = (2)(5) = 10$. Since this limit is nonzero, the quotient rule applies, giving $\lim_{x\to a} h(x) = \frac{-4}{10} = -\frac{2}{5}$.

The quotient rule requires the denominator's limit to be nonzero. In part a, the denominator's limit was the product of two limits, two times five, equaling ten, which is nonzero. Since this condition held, dividing the numerator's limit by the denominator's limit was valid. The answer is that the denominator's limit was nonzero.