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What is the equation of the tangent line of #f(x)=sinx/(x^2-3x+2) # at #x=0#?

Answer 1

#y=1/2x#. See two Socratic graphs, one for zooming the tangent, and the other, for contracting the graph, to reveal branches hidden in the first graph.

graph{(y(x^2-3x+2)-sin x)(y-x/2)=0 [-0.864, 0.764, -0.412, 0.402]} #y = f(0)= 0# at x = 0.

So, the point of contact of the tangent is P( 0, 0 ).

Cross multiply , differentiate and set x = f=0.

#f'(0)(2)+(0)(-3)=cos 0 = 1#, giving the slope #f'(0)=1/2#

And so, the equation of the tangent at P( 0, 0 ) is

#y-0=1/2(x-0)#, giving #y = 1/2x#.

graph{(y(x^2-3x+2)-sin x)(y-x/2)=0 [-26.1, 26, -13.02, 13.02]}

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Answer 2

Evelyn Agard

The equation of the tangent line of f(x)=sinx/(x^2-3x+2) at x=0 is y = -2x + 1.

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Answer from HIX Tutor

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.