document.write( "Question 1027574: Let f(x) = ln(1+x^2). Find the limit as x approaches infinity for f(x), df/dx and df^2/dx^2. Use these limits to explain the graph of the function as x gets large. \n" ); document.write( "

Algebra.Com's Answer #642790 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"f%28x%29+=+ln%281%2Bx%5E2%29\"$ approaches + $\"infinity\"$ as x goes to + $\"infinity\"$ . (Obvious!)\r
\n" ); document.write( "\n" ); document.write( " $\"df%2Fdx+=+%282x%29%2F%281%2Bx%5E2%29\"$ goes to 0 as x goes to + $\"infinity\"$ . (The first derivative is also positive starting at x = 0, hence the graph is increasing as x goes to infinity.)\r
\n" ); document.write( "\n" ); document.write( " $\"d%5E2f%2Fdx%5E2+=+%282-2x%5E2%29%2F%281%2Bx%5E2%29%5E2\"$ goes to 0 as x goes to + $\"infinity\"$ . (The second derivative is negative for x > 0 hence the graph is concave downward there.)\r
\n" ); document.write( "\n" ); document.write( "The preceding information suggest that ln(1+x^2) increases, but slows down significantly as x goes to infinity. (Similar to a 'diminishing returns\" behavior.)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );