Limit of x tends to infinity

To find the limit of a function as x approaches infinity, we need to analyze the behavior of the function as x becomes larger and larger. In this case, we have the function:

f(x) = √(x - sin(x))/(x + cos^2(x))

Simplifying the expression

Before finding the limit, let's simplify the expression to make it more manageable. To do this, we can start by multiplying both the numerator and denominator by the conjugate of the numerator, which is √(x - sin(x)) + √(x + sin(x)):

f(x) = (√(x - sin(x))/(x + cos^2(x))) * (√(x - sin(x)) + √(x + sin(x)))/(√(x - sin(x)) + √(x + sin(x)))

Simplifying further:

f(x) = (x - sin(x) + √(x - sin(x))√(x + sin(x)))/(√(x - sin(x)) + √(x + sin(x)))

Dividing by x

To find the limit as x approaches infinity, we can divide both the numerator and denominator by x:

f(x) = (1 - (sin(x)/x) + (√(x - sin(x))√(x + sin(x)))/x)/((√(x - sin(x)) + √(x + sin(x)))/x)

As x approaches infinity, the terms sin(x)/x and (√(x - sin(x))√(x + sin(x)))/x both tend to zero. This is because sin(x) and x both approach infinity, and dividing them results in a value that approaches zero. Additionally, the square roots in the numerator also approach zero.

Limit as x approaches infinity

After simplifying and dividing by x, the expression becomes:

f(x) = (1 + 0 + 0)/(0 + 0)

Thus, the limit of f(x) as x approaches infinity is undefined.

Summary

The limit of the given function as x approaches infinity is undefined. This is because the terms in the numerator and denominator tend to zero, resulting in an indeterminate form of 0/0. Therefore, we cannot determine a specific value for this limit.