$\lim _{x \rightarrow \infty} \frac{\sqrt{x^2+7 x}}{12-4 x}=$
Answer (1)
Divide both the numerator and the denominator by x .
Rewrite the expression as x 12 − 4 1 + x 7 .
Take the limit as x approaches infinity.
Evaluate the limit to get − 4 1 .
Explanation
Problem Analysis We are asked to compute the limit of a rational function as x approaches infinity. The function is 12 − 4 x x 2 + 7 x .
Strategy To evaluate the limit, we can divide both the numerator and the denominator by x . This will help us simplify the expression and determine the limit as x approaches infinity.
Dividing by x Dividing the numerator and the denominator by x , we get: x 12 − 4 x x x 2 + 7 x = x 12 − 4 x 2 x 2 + 7 x = x 12 − 4 x 2 x 2 + 7 x = x 12 − 4 1 + x 7
Taking the Limit Now, we take the limit as x approaches infinity: x → ∞ lim x 12 − 4 1 + x 7 As x approaches infinity, x 7 approaches 0 and x 12 approaches 0.
Evaluating the Limit So, the limit becomes: 0 − 4 1 + 0 = − 4 1 = − 4 1 = − 4 1
Final Answer Therefore, the limit of the given function as x approaches infinity is − 4 1 .
Examples
In physics, when analyzing the motion of objects or the behavior of systems as they approach extreme conditions (like infinite time or distance), evaluating limits at infinity becomes crucial. For instance, determining the terminal velocity of a falling object involves calculating the limit of its velocity function as time approaches infinity. This helps engineers design structures and systems that can withstand extreme conditions, ensuring safety and efficiency.
