How do you find the limit of $\frac{x + sin x}{x}$ $(x+sinx)/x$ as x approaches 0?

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Answer 1 · 2016-03-03T19:22:56

$2$ $2$

We will make use of the following trigonometric limit:

$lim_(xto0)sinx/x=1$

Let $f(x)=(x+sinx)/x$

Simplify the function:

$f(x)=x/x+sinx/x$

$f(x)=1+sinx/x$

Evaluate the limit:

$lim_(x to 0) (1+sinx/x)$

Split up the limit through addition:

$lim_(x to 0)1+lim_(x to 0)sinx/x$

$1+1=2$

We can check a graph of $(x+sinx)/x$ :

graph{(x+sinx)/x [-5.55, 5.55, -1.664, 3.885]}

The graph does seem to include the point $(0,2)$ , but is in fact undefined.

How do you find the limit of (x+sinx)/xx+sinxx(x+sinx)/x as x approaches 0?