$\lim _{x \rightarrow 3} \sqrt[3]{\frac{x^2+3 x-2}{2 x^2+3 x-2}}$
Answer (1)
Evaluate the numerator x 2 + 3 x − 2 at x = 3 , which gives 16.
Evaluate the denominator 2 x 2 + 3 x − 2 at x = 3 , which gives 25.
Substitute the values into the function, obtaining 3 25 16 .
Calculate the cube root to find the limit: 0.8618 .
Explanation
Problem Analysis and Setup We are asked to find the limit of the cube root of a rational function as x approaches 3. The function is given by: f ( x ) = 3 2 x 2 + 3 x − 2 x 2 + 3 x − 2
To find the limit, we can first check if we can directly substitute x = 3 into the function.
Evaluating Numerator and Denominator Let's evaluate the numerator and the denominator separately at x = 3 :
Numerator: x 2 + 3 x − 2 = ( 3 ) 2 + 3 ( 3 ) − 2 = 9 + 9 − 2 = 16
Denominator: 2 x 2 + 3 x − 2 = 2 ( 3 ) 2 + 3 ( 3 ) − 2 = 2 ( 9 ) + 9 − 2 = 18 + 9 − 2 = 25
Since the denominator is not zero at x = 3 , we can substitute x = 3 directly into the function.
Substituting into the Function Now, we can substitute these values into the function:
3 25 16
We can calculate the cube root of this fraction.
Calculating the Cube Root 3 25 16 = ( 25 16 ) 3 1 ≈ 0.8618
Final Answer Therefore, the limit of the function as x approaches 3 is approximately 0.8618.
x → 3 lim 3 2 x 2 + 3 x − 2 x 2 + 3 x − 2 = 3 25 16 ≈ 0.8618
Examples
In engineering, when analyzing the stability of a system as a parameter approaches a certain value, you might encounter limits of rational functions. For example, the gain of a control system might be described by a function similar to the one in this problem. Evaluating the limit as a parameter approaches a critical value helps engineers understand the system's behavior near that critical point, ensuring the system remains stable and performs as expected. This is crucial in designing robust and reliable control systems.
