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

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Answer 1 · 2016-03-03T19:22:56

#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.