document.write( "Question 1200106: Find all horizontal and vertical asymptotes of the function: f(x) = x/sqrt(x^2-1)
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Algebra.Com's Answer #834146 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "The denominator is $\"sqrt%28x%5E2-1%29\"$ , so the radicand $\"x%5E2-1\"$ must be non-negative.

\n" ); document.write( "Furthermore, since that square root is in the denominator,the radicand can't be zero.

\n" ); document.write( " $\"x%5E2-1=%28x%2B1%29%28x-1%29\"$ , so the function is undefined on [-1,1].

\n" ); document.write( "For x values a tiny bit greater than 1, the denominator is close to 0, and the numerator is positive, so the function value is large positive. That makes x=1 a vertical asymptote.

\n" ); document.write( "Similarly, for x values a tiny bit less than -1, the denominator is close to 0, and here the numerator is negative, so the function value is large negative. So x=-1 is also a vertical asymptote.

\n" ); document.write( "For either large positive or large negative values of x, $\"sqrt%28x%5E2-1%29\"$ is very close to $\"sqrt%28x%5E2%29=abs%28x%29\"$ , so the function is very nearly equal to $\"x%2Fabs%28x%29\"$ .

\n" ); document.write( "For large positive values of x, $\"x%2Fabs%28x%29=1\"$ , so y=1 is a horizontal asymptote for x>1.

\n" ); document.write( "For large negative values of x, $\"x%2Fabs%28x%29=-1\"$ , so y=-1 is a horizontal asymptote for x<-1.

\n" ); document.write( "ANSWERS:
\n" ); document.write( "vertical asymptotes at x=-1 and x=1;
\n" ); document.write( "horizontal asymptotes at y=1 (for positive x values) and y=-1 (for negative x values).

\n" ); document.write( "A graph of the function and the two horizontal asymptotes....

\n" ); document.write( " $\"graph%28400%2C400%2C-3%2C3%2C-3%2C3%2Cx%2Fsqrt%28x%5E2-1%29%2C-1%2C1%29\"$
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