find lim f(x) - lim f(x)
x->infinity x->-infinity
if f(x)=ln (x+sqrt[x^2+a^2])/(x+sqrt[x^2+b^2])
a and b are real numbers
To find the limits of the given function, we need to evaluate the limits as x approaches positive infinity and negative infinity separately.
1. Limit as x approaches positive infinity:
For this limit, we need to consider the dominant terms as x becomes very large.
First, let's simplify the function by removing the square roots:
f(x) = ln[(x + sqrt(x^2 + a^2))/(x + sqrt(x^2 + b^2))]
As x approaches positive infinity, the square root terms inside the logarithm become negligible compared to x. So we can simplify further:
f(x) ≈ ln[(x + x)/(x + x)] = ln(2)
Therefore, the limit as x approaches positive infinity is ln(2).
2. Limit as x approaches negative infinity:
Similarly, consider the dominant terms as x becomes very negative.
Using the same simplification:
f(x) = ln[(x + sqrt(x^2 + a^2))/(x + sqrt(x^2 + b^2))]
As x approaches negative infinity, the square root terms inside the logarithm become negligible compared to x. So we can simplify further:
f(x) ≈ ln[(-x + -x)/(-x + -x)] = ln(2)
Therefore, the limit as x approaches negative infinity is also ln(2).
So the final answer is:
lim f(x) - lim f(x) = ln(2) - ln(2) = 0