# How do you find the points where the graph of the function #y= x +sinx# has horizontal tangents?

The tangent line is horizontal if and only if the slope of the tangetn line is

To help form an image of the solution above, here is the graph of the function. (You can zoom in/out and drag the graph around to explore it a bit.)

graph{x+sinx [-25.52, 25.82, -15.62, 10.04]}

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To find the points where the graph of the function y = x + sin(x) has horizontal tangents, we need to determine the values of x where the derivative of the function is equal to zero.

First, let's find the derivative of the function y = x + sin(x). The derivative of x with respect to x is 1, and the derivative of sin(x) with respect to x is cos(x). Therefore, the derivative of y with respect to x is 1 + cos(x).

To find the points where the derivative is equal to zero, we set 1 + cos(x) equal to zero and solve for x.

1 + cos(x) = 0

cos(x) = -1

x = π + 2πn, where n is an integer.

So, the points where the graph of y = x + sin(x) has horizontal tangents are x = π + 2πn, where n is an integer.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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