\(\lim _{x \rightarrow-\infty} \frac{7 x-3}{x^3+2 x-9}=\)
Answer (1)
Divide both numerator and denominator by x 3 . This gives 1 + x 2 2 − x 3 9 x 2 7 − x 3 3 .
As x → − ∞ , the terms x 2 7 , x 3 3 , x 2 2 , and x 3 9 all approach 0.
The expression simplifies to 1 + 0 − 0 0 − 0 = 1 0 .
Thus, the limit is 0 .
Explanation
Problem Analysis We are asked to find the limit of the given rational function as x approaches − ∞ . The function is:
f ( x ) = x 3 + 2 x − 9 7 x − 3
To find this limit, we can divide both the numerator and the denominator by the highest power of x that appears in the denominator, which is x 3 . This will help us analyze the behavior of the function as x becomes very large in the negative direction.
Dividing by the Highest Power Dividing both the numerator and the denominator by x 3 , we get:
x → − ∞ lim x 3 x 3 + x 3 2 x − x 3 9 x 3 7 x − x 3 3 = x → − ∞ lim 1 + x 2 2 − x 3 9 x 2 7 − x 3 3
Now, we analyze what happens to each term as x approaches − ∞ .
Evaluating the Limit As x approaches − ∞ :
x 2 7 approaches 0 because the denominator grows without bound while the numerator remains constant.
x 3 3 approaches 0 for the same reason.
x 2 2 approaches 0.
x 3 9 approaches 0.
Thus, the expression simplifies to:
x → − ∞ lim 1 + 0 − 0 0 − 0 = 1 0 = 0
Therefore, the limit of the function as x approaches − ∞ is 0.
Final Answer The limit of the function as x approaches − ∞ is 0.
x → − ∞ lim x 3 + 2 x − 9 7 x − 3 = 0
Examples
In physics, when analyzing the behavior of systems at extreme distances or times, we often encounter limits at infinity. For example, when studying the gravitational potential of an object as you move infinitely far away, the potential approaches zero. This concept is crucial in understanding long-range interactions and the overall structure of physical systems.
