Mister Exam
Graphing y = (cot(sqrt(x)))^cos(x)
The solution
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cos(x)/ ___\ f(x) = cot \\/ x /
$$f{\left(x \right)} = \cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)}$$
f = cot(sqrt(x))^cos(x)
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 9.86960440108936$$
so we need to solve the equation:
$$\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 9.86960440108936$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = \left(- i\right)^{\left\langle -1, 1\right\rangle}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left(- i\right)^{\left\langle -1, 1\right\rangle}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)}$$
$$\lim_{x \to -\infty} \cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = \left(- i\right)^{\left\langle -1, 1\right\rangle}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left(- i\right)^{\left\langle -1, 1\right\rangle}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(sqrt(x))^cos(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)}}{x}\right)$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = \cot^{\cos{\left(x \right)}}{\left(\sqrt{- x} \right)}$$
- No
$$\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = - \cot^{\cos{\left(x \right)}}{\left(\sqrt{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = \cot^{\cos{\left(x \right)}}{\left(\sqrt{- x} \right)}$$
- No
$$\cot^{\cos{\left(x \right)}}{\left(\sqrt{x} \right)} = - \cot^{\cos{\left(x \right)}}{\left(\sqrt{- x} \right)}$$
- No
so, the function
not is
neither even, nor odd