$\lim _{x \rightarrow \infty} \frac{-14 x}{12+2 x}=$
Answer (1)
Divide both numerator and denominator by x : lim x → ∞ 12 + 2 x − 14 x = lim x → ∞ x 12 + 2 − 14 .
As x approaches infinity, x 12 approaches 0.
Simplify the expression: 0 + 2 − 14 = − 7 .
The limit of the function as x approaches infinity is − 7 .
Explanation
Problem Analysis We are asked to find the limit of the function 12 + 2 x − 14 x as x approaches infinity. This is a limit of a rational function as x goes to infinity.
Divide by the Highest Power of x To find the limit of a rational function as x approaches infinity, we can divide both the numerator and the denominator by the highest power of x that appears in the denominator. In this case, the highest power of x in the denominator is x .
Simplify the Expression Dividing both the numerator and the denominator by x , we get: x → ∞ lim 12 + 2 x − 14 x = x → ∞ lim x 12 + x 2 x x − 14 x = x → ∞ lim x 12 + 2 − 14
Evaluate the Limit Now, as x approaches infinity, the term x 12 approaches 0. Therefore, we have: x → ∞ lim x 12 + 2 − 14 = 0 + 2 − 14 = 2 − 14 = − 7
Final Answer Thus, the limit of the given function as x approaches infinity is -7.
Examples
In electrical engineering, when analyzing circuits with resistors, the behavior of current in a complex circuit as resistance approaches infinity can be modeled using limits. Similarly, in physics, understanding the terminal velocity of an object falling through a fluid involves evaluating a limit as time approaches infinity. These examples demonstrate how limits help predict the long-term behavior of systems.
