91Ó°ÊÓ

When you need to find the limit of a sum of two functions, you can use a simple rule in calculus: the limit of a sum is equal to the sum of the limits of each function. Suppose you have two functions, `f(x)` and `g(x)`, and their limits as `x` approaches a specific value are known. Using the given problem, we have:o `\( \lim_{x \rightarrow 0} f(x) = -\frac{1}{2} \)o `\( \lim_{x \rightarrow 0} g(x) = \frac{1}{2} \)To find `\( \lim_{x \rightarrow 0} (f(x) + g(x)) \)`, we sum the individual limits:```\[ \lim_{x \rightarrow 0} (f(x) + g(x)) = \lim_{x \rightarrow 0} f(x) + \lim_{x \rightarrow 0} g(x) \]\[ = -\frac{1}{2} + \frac{1}{2} = 0 \]```So, the limit of `\( f(x) + g(x) \)` as `x` approaches 0 is 0.

Finding the limit of a difference of two functions follows a similar principle to that of sums. The limit of the difference is equal to the difference of the limits. Given the same functions `f(x)` and `g(x)` with known limits:o `\( \lim_{x \rightarrow 0} f(x) = -\frac{1}{2} \)o `\( \lim_{x \rightarrow 0} g(x) = \frac{1}{2} \)To find the limit of `\( f(x) - 2g(x) \)`, you need to first multiply the limit of `g(x)` by 2 and then subtract it from the limit of `f(x)`:```\[ \lim_{x \rightarrow 0} (f(x) - 2g(x)) = \lim_{x \rightarrow 0} f(x) - 2 \lim_{x \rightarrow 0} g(x) \]\[ = -\frac{1}{2} - 2 \left( \frac{1}{2} \right) = -\frac{1}{2} - 1 = -\frac{3}{2} \]```Thus, the limit of `\( f(x) - 2g(x) \)` as `x` approaches 0 is `-\frac{3}{2}`.

Calculating the limit of a product of two functions involves another straightforward rule: the limit of a product is the product of the limits. With the same given limits for `f(x)` and `g(x)`, we apply this rule as follows:```\[ \lim_{x \rightarrow 0} (f(x) \cdot g(x)) = \lim_{x \rightarrow 0} f(x) \cdot \lim_{x \rightarrow 0} g(x) \]\[ = \left( -\frac{1}{2} \right) \cdot \left( \frac{1}{2} \right) = -\frac{1}{4} \]```So, the limit of `\( f(x) \cdot g(x) \)` as `x` approaches 0 is `-\frac{1}{4}`.

Lastly, to find the limit of a quotient of two functions, another important rule in calculus applies: the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. Using the provided limits for `f(x)` and `g(x)`:```\[ \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow 0} f(x)}{\lim_{x \rightarrow 0} g(x)} \]\[ = \frac{-\frac{1}{2}}{\frac{1}{2}} = -1 \]```So, the limit of `\( \frac{f(x)}{g(x)} \)` as `x` approaches 0 is `-1`. Remember, this rule can only be used if the limit of the denominator is not zero, which in this case, it is `\frac{1}{2}`.