Mister Exam
Graphing y = 5*sin(5x)+9*sin(9x)
The solution
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f(x) = 5*sin(5*x) + 9*sin(9*x)
$$f{\left(x \right)} = 5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}$$
f = 5*sin(5*x) + 9*sin(9*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
$$\lim_{x \to -\infty}\left(5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -14, 14\right\rangle$$
$$\lim_{x \to \infty}\left(5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -14, 14\right\rangle$$
$$\lim_{x \to -\infty}\left(5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -14, 14\right\rangle$$
$$\lim_{x \to \infty}\left(5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -14, 14\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 5*sin(5*x) + 9*sin(9*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)} = - 5 \sin{\left(5 x \right)} - 9 \sin{\left(9 x \right)}$$
- No
$$5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)} = 5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)} = - 5 \sin{\left(5 x \right)} - 9 \sin{\left(9 x \right)}$$
- No
$$5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)} = 5 \sin{\left(5 x \right)} + 9 \sin{\left(9 x \right)}$$
- Yes
so, the function
is
odd