SOLUTION: 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.

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.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
approaches + as x goes to +. (Obvious!)
goes to 0 as x goes to +. (The first derivative is also positive starting at x = 0, hence the graph is increasing as x goes to infinity.)
goes to 0 as x goes to +. (The second derivative is negative for x > 0 hence the graph is concave downward there.)
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.)

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