# For #f(t)= (1/(t-3),t^2)# what is the distance between #f(0)# and #f(2)#?

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To find the distance between ( f(0) ) and ( f(2) ) for ( f(t) = \left( \frac{1}{{t - 3}}, t^2 \right) ), evaluate ( f(0) ) and ( f(2) ), then use the distance formula in two dimensions.

First, ( f(0) ) is ( \left( \frac{1}{-3}, 0 \right) ), which simplifies to ( \left( -\frac{1}{3}, 0 \right) ).

Next, ( f(2) ) is ( \left( \frac{1}{2 - 3}, 2^2 \right) ), which simplifies to ( \left( 1, 4 \right) ).

Now, apply the distance formula ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ) to find the distance between these points.

Substitute the coordinates into the formula: ( d = \sqrt{(1 - (-\frac{1}{3}))^2 + (4 - 0)^2} ).

Simplify: ( d = \sqrt{\left( \frac{4}{3} \right)^2 + 16} = \sqrt{\frac{16}{9} + 16} = \sqrt{\frac{16}{9} + \frac{144}{9}} = \sqrt{\frac{160}{9}} ).

Finally, ( d = \frac{4\sqrt{10}}{3} ).

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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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