document.write( "Question 1209595: limit x (1 - 2 cos ((pi x)/(3 x - sqrt3))) as x \[LongRightArrow] + \[Infinity] \n" ); document.write( "
Algebra.Com's Answer #849673 by CPhill(1959)\"\" \"About 
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Here's how to evaluate the limit:\r
\n" ); document.write( "\n" ); document.write( "**1. Simplify the argument of the cosine function:**\r
\n" ); document.write( "\n" ); document.write( "As x approaches infinity, the term -√3 in the denominator becomes insignificant compared to 3x. So, we can approximate the argument of the cosine as:\r
\n" ); document.write( "\n" ); document.write( "(πx) / (3x - √3) ≈ (πx) / (3x) = π/3\r
\n" ); document.write( "\n" ); document.write( "**2. Evaluate the limit:**\r
\n" ); document.write( "\n" ); document.write( "Now, substitute this approximation back into the original expression:\r
\n" ); document.write( "\n" ); document.write( "lim (x→∞) [1 - 2cos(πx / (3x - √3))] = 1 - 2cos(π/3)\r
\n" ); document.write( "\n" ); document.write( "Since cos(π/3) = 1/2, we have:\r
\n" ); document.write( "\n" ); document.write( "1 - 2 * (1/2) = 1 - 1 = 0\r
\n" ); document.write( "\n" ); document.write( "Therefore, the limit is 0.
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