Mister Exam
Graphing y = sinx+x^2021
The solution
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2021 f(x) = sin(x) + x
$$f{\left(x \right)} = x^{2021} + \sin{\left(x \right)}$$
f = x^2021 + sin(x)
The graph of the function
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x^{2021} + \sin{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2021} + \sin{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x^{2021} + \sin{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x^{2021} + \sin{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x) + x^2021, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{2021} + \sin{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x^{2021} + \sin{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{x^{2021} + \sin{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x^{2021} + \sin{\left(x \right)}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$x^{2021} + \sin{\left(x \right)} = - x^{2021} - \sin{\left(x \right)}$$
- No
$$x^{2021} + \sin{\left(x \right)} = x^{2021} + \sin{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x^{2021} + \sin{\left(x \right)} = - x^{2021} - \sin{\left(x \right)}$$
- No
$$x^{2021} + \sin{\left(x \right)} = x^{2021} + \sin{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd