91Ó°ÊÓ

Limit computation involves finding the value that a function approaches as its input approaches a particular point. In this example, as the variable \( x \) approaches 1, we are interested in determining the value of the expression \( \lim _{x \rightarrow 1}\left[\frac{f(x)}{g(x)-h(x)}\right] \).
Understanding limits is crucial because they form the foundation for defining derivatives and integrals in calculus.
To compute limits, we can use multiple predefined limit laws, which make the process straightforward:

• Substitution: Directly substitute the value into the function, if the function is continuous at that point.
• Arithmetic Operations: Knowing the limits of individual functions allows for their sum, difference, product, and quotient to be calculated.
• Simplifying the Expression: If direct substitution leads to an indeterminate form, such as 0/0, the expression may need to be simplified first.

This exercise combines these approaches, leading us smoothly to the solution.

The quotient rule is a key limit law used when calculating the limit of a quotient of two functions. It states that, given two functions \( f(x) \) and \( g(x) \) where the limit of \( g(x) eq 0 \), you can find:\[ \lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)} \]In our exercise, the quotient rule allows us to separate the limit of a division into a simpler problem by evaluating the limits of its numerator and denominator separately.
We start by applying the quotient rule:

• Find the limit of the numerator, \( \lim_{x \rightarrow 1} f(x) = 8 \).
• Find the limit of the denominator, \( \lim_{x \rightarrow 1} (g(x) - h(x)) \), using the difference rule.

This rule clarifies that the limit computation is valid as long as the limit of the denominator is non-zero, ensuring the division is permissible.

The difference rule is applied to simplify the computation of the limit of a difference of two functions. It states that:\[ \lim_{x \rightarrow a} (u(x) - v(x)) = \lim_{x \rightarrow a} u(x) - \lim_{x \rightarrow a} v(x) \]In the given problem, we use the difference rule to find:\[ \lim_{x \rightarrow 1} (g(x) - h(x)) = \lim_{x \rightarrow 1} g(x) - \lim_{x \rightarrow 1} h(x) = 3 - 2 \]This simplifies directly to 1, showing that the denominator of our fraction remains consistent and non-zero.
Thus ensuring we can apply the Quotient Rule without complication.
The difference rule is crucial because it allows us to evaluate complex expressions by breaking them into simpler parts, fundamentally aiding in finding the limit of complicated functions.