Find the limit, if it exists.
$\lim _{x \rightarrow \infty} \frac{7}{2 x+\sin (x)}$
Answer (2)
Analyze the behavior of the denominator 2 x + sin ( x ) as x approaches infinity.
Recognize that the denominator approaches infinity while the numerator remains constant.
Apply the limit rule: lim x → ∞ f ( x ) c = 0 if c is a constant and f ( x ) approaches infinity.
Conclude that the limit is 0 .
Explanation
Problem Analysis We are asked to find the limit of the function 2 x + s i n ( x ) 7 as x approaches infinity.
Analyzing the Denominator As x approaches infinity, 2 x also approaches infinity. The sine function, sin ( x ) , oscillates between -1 and 1. Therefore, the denominator 2 x + sin ( x ) will also approach infinity.
Determining the Limit Since the numerator is a constant (7) and the denominator approaches infinity, the limit of the fraction is 0.
Final Answer Therefore, lim x → ∞ 2 x + s i n ( x ) 7 = 0 .
Examples
In electrical engineering, when analyzing the behavior of circuits with alternating current (AC) sources, understanding limits at infinity helps determine the steady-state response of the circuit. For example, the function might represent the voltage across a capacitor as time approaches infinity. Knowing that the limit is zero indicates that the voltage stabilizes to zero over a long period, which is crucial for designing stable and reliable circuits.
The limit as x approaches infinity for the function 2 x + s i n ( x ) 7 is zero. This is because the denominator grows indefinitely while the numerator remains a constant value. Applying the limit rule, the final result is 0 .
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