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

Answer 1

Manipulate the fundamental trigonometric limit to get #lim_(x->0)x/sinx=1#.

Evaluating the limit directly produces the indeterminate form #0/0# so we'll have to use an alternate method.
We know that #lim_(x->0)sinx/x=1#, so we'll work from there.
Note that #x/sinx=1/(sinx/x)#, which means #lim_(x->0)x/sinx# can be equivalently expressed as #lim_(x->0)1/(sinx/x)#. Using the properties of limits, this becomes: #(lim_(x->0)1)/(lim_(x->0)sinx/x#
Well, #lim_(x->0)1=1# for all #x#, and #lim_(x->0)sinx/x=1# (fundamental trigonometric limit). That means we have: #1/1#
Which is simply #1#.
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Answer 2

To find the limit of x/sin(x) as x approaches 0, 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. The derivative of x with respect to x is 1, and the derivative of sin(x) with respect to x is cos(x).

Therefore, the limit of x/sin(x) as x approaches 0 is equal to the limit of 1/cos(x) as x approaches 0.

Now, substituting x = 0 into the expression 1/cos(x), we get 1/cos(0) = 1/1 = 1.

Hence, the limit of x/sin(x) as x approaches 0 is equal to 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.

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.

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.

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.

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