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How do you find the limit as x approaches 1 of #sin(pix)/(x-1)#?

Answer 1

I would use the de l'Hospital Rule. When your limit gives you #0/0# you can derive the numerator and denominator and then do the limit. In your case you get: #lim_(x->1)[(pi*cos(pix))/1]=-pi#

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Answer 2

Emma Aldridge

To find the limit as x approaches 1 of sin(pix)/(x-1), we can use the concept of L'Hôpital's Rule. By applying this rule, we differentiate both the numerator and denominator with respect to x.

Differentiating the numerator, we get cos(pix) * pi. Differentiating the denominator, we get 1.

Now, we can evaluate the limit by substituting x = 1 into the differentiated expressions.

The numerator becomes cos(pi) * pi, which simplifies to -pi. The denominator remains as 1.

Therefore, the limit as x approaches 1 of sin(pix)/(x-1) is -pi.

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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.