lim
x → ∞
tan−1(x7 − x9)
You must have an idea what the tangent function looks like. As Ø approaches 90° or π/2 from the left, tanØ becomes infinitely large positively.
as Ø approaches 90° from the right, it is infinitely negative
now look at (x^7 - x^9), as x --> ∞ , it is hugely negative
thus tan^-1 (hugely negatiave ) approaches 90° or π/2
of course another answer would be -90° or -π/2
To evaluate the limit as x approaches infinity of tan^(-1)(x^7 - x^9), we can first simplify the expression inside the tangent function.
Step 1: Rewrite the expression.
tan^(-1)(x^7 - x^9) = tan^(-1)(x^7(1 - x^2))
Step 2: Use the identity tan^(-1)(a) = pi/2 - tan^(-1)(1/a) to simplify further.
tan^(-1)(x^7(1 - x^2)) = pi/2 - tan^(-1)(1/[x^7(1 - x^2)])
Step 3: Consider the limit as x approaches infinity.
As x approaches infinity, both x^7 and (1 - x^2) go to infinity. Therefore, we have 1/[x^7(1 - x^2)] approaching zero.
Step 4: Simplify the expression.
tan^(-1)(1/[x^7(1 - x^2)]) approaches tan^(-1)(0), which equals 0.
Step 5: Use the identity tan^(-1)(a) = pi/2 - tan^(-1)(1/a) to find the limit.
Therefore, the limit as x approaches infinity of tan^(-1)(x^7 - x^9) is equal to pi/2 - 0, which is pi/2.
To find the limit of the given expression as x approaches infinity, we can determine the behavior of the function as x becomes larger.
Let's break down the expression: tan^(-1)(x^7 - x^9).
As x approaches infinity, the term (x^7 - x^9) becomes dominated by the highest power of x, which in this case is x^9. This means that the term x^9 is the most significant term in the expression.
When x^9 becomes much larger, the x^7 term becomes negligible. So, we can approximate the expression as tan^(-1)(-x^9) as x approaches infinity.
Next, we need to analyze the behavior of the arctangent function. The arctangent function is limited to a range of (-π/2, π/2). As the argument of the arctangent function becomes more negative, the value approaches -π/2 (-90 degrees).
Therefore, as x approaches infinity, the expression tan^(-1)(-x^9) approaches -π/2.
In summary, the limit of the given expression as x approaches infinity is -π/2 or approximately -1.5708 radians.