$\lim _{x \rightarrow-\infty} \frac{\sqrt{x^2+7 x}}{12-4 x}=$
Answer (1)
Divide both numerator and denominator by x , considering that x 2 = − x for x < 0 .
Simplify the expression to x 12 − 4 − 1 + x 7 .
Evaluate the limit as x → − ∞ , noting that x 7 and x 12 approach 0.
The limit evaluates to − 4 − 1 , so the final answer is 4 1 .
Explanation
Problem Setup We are asked to find the limit of the expression 12 − 4 x x 2 + 7 x as x approaches − ∞ .
Simplifying the Expression To evaluate this limit, we need to consider the behavior of the function as x becomes a very large negative number. We can divide both the numerator and the denominator by x to simplify the expression.
Accounting for Negative x Since x is approaching − ∞ , x is negative. Therefore, when we divide the square root by x , we need to account for the fact that x 2 = ∣ x ∣ = − x when x < 0 . So, we have 12 − 4 x x 2 + 7 x = 12 − 4 x x 2 ( 1 + x 7 ) = 12 − 4 x ∣ x ∣ 1 + x 7 = 12 − 4 x − x 1 + x 7 Now, divide both the numerator and the denominator by x :
12 − 4 x − x 1 + x 7 = x 12 − 4 − 1 + x 7
Evaluating the Limit Now we can take the limit as x approaches − ∞ :
x → − ∞ lim x 12 − 4 − 1 + x 7 As x → − ∞ , x 7 → 0 and x 12 → 0 . Therefore, x → − ∞ lim x 12 − 4 − 1 + x 7 = 0 − 4 − 1 + 0 = − 4 − 1 = 4 1
Final Answer Thus, the limit of the given expression as x approaches − ∞ is 4 1 .
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, when calculating the gravitational potential far from a massive object, we might use limits to determine the potential's asymptotic behavior. Understanding how functions behave as their inputs approach infinity or negative infinity is crucial for making accurate predictions and simplifying complex models.
