Let [tex] \lim _{x \rightarrow 2} f(x)=7 [/tex] and [tex] \lim _{x \rightarrow 2} g(x)=14 [/tex]. Use the limit rules to find the following limit.
[tex] \lim _{x \rightarrow 2} \frac{f(x)}{g(x)} [/tex]
[tex] \lim _{x \rightarrow 2} \frac{f(x)}{g(x)}= [/tex]
(Type an integer or a simplified fraction.)
Answer (1)
Apply the quotient rule for limits: lim x → 2 g ( x ) f ( x ) = l i m x → 2 g ( x ) l i m x → 2 f ( x ) .
Substitute the given limits: lim x → 2 g ( x ) f ( x ) = 14 7 .
Simplify the fraction: 14 7 = 2 1 .
The limit is 2 1 .
Explanation
Problem Analysis We are given that lim x → 2 f ( x ) = 7 and lim x → 2 g ( x ) = 14 . We want to find lim x → 2 g ( x ) f ( x ) .
Applying Limit Rule We can use the limit rule that states the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. In this case, we have:
Quotient of Limits x → 2 lim g ( x ) f ( x ) = lim x → 2 g ( x ) lim x → 2 f ( x )
Substitution Substituting the given values, we get:
Calculating the Limit x → 2 lim g ( x ) f ( x ) = 14 7
Simplifying the Fraction Simplifying the fraction, we have:
Final Simplification 14 7 = 2 1
Examples
Imagine you are analyzing the behavior of two stocks, f(x) and g(x), as time approaches a certain day (x=2). f(x) represents the profit of one stock, and g(x) represents the profit of another stock. Knowing that the limit of f(x) is 7 and the limit of g(x) is 14, you can determine the limit of the ratio f(x)/g(x), which represents the relative performance of the two stocks as x approaches 2. This can help you make informed decisions about which stock to invest in.
