Evaluate the limit
$\lim _{x \rightarrow \infty} \frac{\sqrt{8+3 x^2}}{(2+4 x)}$
Answer (1)
Divide both the numerator and the denominator by x .
Rewrite the expression as lim x → ∞ x 2 + 4 x 2 8 + 3 .
Evaluate the limit by taking x → ∞ , which gives 4 3 .
The final answer is 4 3 .
Explanation
Problem Analysis We are asked to evaluate the limit of a rational function as x approaches infinity. The function is given by x → ∞ lim ( 2 + 4 x ) 8 + 3 x 2 To solve this, we need to analyze the behavior of the function as x becomes very large.
Dividing by x 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.
x → ∞ lim x 2 + 4 x x 8 + 3 x 2 Since x is approaching infinity, we can assume 0"> x > 0 . Therefore, x = x 2 . We can rewrite the numerator as:
x 8 + 3 x 2 = x 2 8 + 3 x 2 = x 2 8 + 3 x 2 = x 2 8 + 3 And the denominator as:
x 2 + 4 x = x 2 + 4 Now, we can rewrite the limit as:
x → ∞ lim x 2 + 4 x 2 8 + 3
Evaluating the Limit Now, we evaluate the limit as x approaches infinity. As x → ∞ , we have:
x 2 8 → 0 and
x 2 → 0 So the limit becomes:
0 + 4 0 + 3 = 4 3 Thus, the limit is 4 3 .
Final Answer Therefore, the limit of the given function as x approaches infinity is 4 3 .
x → ∞ lim ( 2 + 4 x ) 8 + 3 x 2 = 4 3 We can approximate this value as:
4 3 ≈ 0.433
Examples
In physics, when analyzing the motion of objects or the behavior of systems as time or distance approaches infinity, evaluating limits like this helps to understand the long-term behavior or stability of the system. For example, determining the terminal velocity of an object falling through a fluid involves evaluating a limit as time approaches infinity. This concept is also crucial in engineering for designing stable control systems and predicting their performance over extended periods.
