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.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> 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.       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+ln%281%2Bx%5E2%29 approaches +infinity as x goes to +infinity. (Obvious!)
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.)
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.)
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.)