Find the limit, if it exists. (If an answer does not exist, enter DNE.)
$\lim _{x \rightarrow-\infty} \frac{6 x^2}{x+5}$
Answer (1)
Divide both numerator and denominator by x : lim x → − ∞ x + 5 6 x 2 = lim x → − ∞ 1 + x 5 6 x .
As x → − ∞ , x 5 approaches 0: lim x → − ∞ 1 + 0 6 x = lim x → − ∞ 6 x .
Evaluate the limit: As x → − ∞ , 6 x approaches − ∞ .
The limit does not exist: D NE .
Explanation
Problem Analysis We are asked to find the limit of the function x + 5 6 x 2 as x approaches − ∞ . This is a limit of a rational function as x goes to infinity.
Dividing by x To evaluate the limit, we can divide both the numerator and the denominator by the highest power of x that appears in the denominator, which is x . This gives us: x → − ∞ lim x + 5 6 x 2 = x → − ∞ lim x x + x 5 x 6 x 2 = x → − ∞ lim 1 + x 5 6 x
Evaluating the Limit As x approaches − ∞ , the term x 5 approaches 0. Therefore, we have: x → − ∞ lim 1 + x 5 6 x = x → − ∞ lim 1 + 0 6 x = x → − ∞ lim 6 x
Final Result As x approaches − ∞ , 6 x also approaches − ∞ . Therefore, the limit is − ∞ . Since the limit is infinite, we say that the limit does not exist.
Examples
In physics, when analyzing the motion of objects or the behavior of fields at extreme distances, we often encounter limits at infinity. For example, understanding how the gravitational force between two objects behaves as the distance between them becomes infinitely large involves evaluating a limit similar to the one in this problem. This helps physicists make predictions and understand the fundamental laws governing the universe.
