Determine $\lim _{x \rightarrow-2} 3 x+x^2$.
a) 6
b) -4
c) 10
d) -2
Answer (2)
We are asked to find the limit of the function f ( x ) = 3 x + x 2 as x approaches − 2 .
Substitute x = − 2 into the function: f ( − 2 ) = 3 ( − 2 ) + ( − 2 ) 2 .
Calculate the value: 3 ( − 2 ) + ( − 2 ) 2 = − 6 + 4 = − 2 .
The limit of the function as x approaches − 2 is − 2 .
Explanation
Problem Analysis We are asked to find the limit of the function f ( x ) = 3 x + x 2 as x approaches − 2 . Since this is a polynomial function, we can evaluate the limit by direct substitution.
Substitution Substitute x = − 2 into the function: f ( − 2 ) = 3 ( − 2 ) + ( − 2 ) 2 .
Calculation Calculate the value: 3 ( − 2 ) + ( − 2 ) 2 = − 6 + 4 = − 2 .
Final Answer Therefore, the limit of the function as x approaches − 2 is − 2 .
Examples
In physics, when analyzing the motion of an object, you might encounter polynomial functions describing its position over time. Determining the limit as time approaches a specific value helps predict the object's position at that instant. For example, if the position of a particle is given by s ( t ) = 3 t + t 2 , finding lim t → − 2 s ( t ) would tell us the particle's theoretical position at t = − 2 , even if the time value is physically not meaningful in the context of the problem. This concept is crucial for understanding the behavior of systems as they approach certain conditions.
The limit of the function 3 x + x 2 as x approaches − 2 is − 2 . Therefore, the correct answer is option (d).
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