Question

How do you find the limit of `(sin(x+pi)) / x` as x approaches 0?

Answer 1

`-1`

Explanation:

The limit of the ratio is the limit of the ratio of the derivatives of the numerator and denominator.

`lim x to 0 sin(x+pi)/x= lim x to 0 cos(x+pi)/1 = cos pi /1 = -1.`

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Answer 2

Rewrite `sin(x+pi)` as `-sinx`, then use the fundamental trigonometric limit.

Explanation:

`sin(A+B) = sinAcosB+cosAsinB`

`sin(x+pi) = sinxcospi+cosxsinpi = sinx(-1)+cosx(0) = -sinx`

So,

`lim_(xrarr0) sin(x+pi)/x = lim_(xrarr0)(-sinx)/x`

`= -1 lim_(xrarr0)(sinx)/x`

`= -1(1) = -1`

(using `lim_(x rarr 0) sin x/ x = 1`)