How do you find #lim (t+1/t)((4-t)^(3/2)-8)# as #t->0# using l'Hospital's Rule?

Answer 1

Please see below.

#(t+1/t)((4-t)^(3/2)-8)# as #trarr0# has form #oo*0#.
We will rewrite it as #((4-t)^(3/2)-8)/(1/(t+1/t)# so we have an expression whose limit has form #0/0# and we can use l'Hospital's Rule.
Note that #t+1/t = (t^2+1)/t# so we want
#lim_(trarro)(t+1/t)((4-t)^(3/2)-8) = lim+(trarr0)((4-t)^(3/2)-8)/(1/(t+1/t)#
# = lim_(trarr0)((4-t)^(3/2)-8)/(t/(t^2+1))#
# = lim_(trarr0)(-(3/2)(4-t)^(1/2))/((1(t^2+1)-t(2t))/(t^2+1)^2)#
# = lim_(trarr0)(-(3/2)(4-t)^(1/2))/((1-t^2)/(t^2+1)^2)#
# = ((-3/2)(2))/(1/1^2) = -3#
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Answer 2

To find the limit ( \lim_{t \to 0} (t + \frac{1}{t})((4 - t)^{\frac{3}{2}} - 8) ) using L'Hôpital's Rule, we first rewrite the expression to an indeterminate form ( \frac{0}{0} ), then take the derivative of the numerator and the derivative of the denominator separately.

  1. Rewrite the expression:

[ \lim_{t \to 0} (t + \frac{1}{t})((4 - t)^{\frac{3}{2}} - 8) ]

  1. Expand the expression:

[ = \lim_{t \to 0} \frac{(t + \frac{1}{t})(\sqrt{4 - t} - 2\sqrt{2})(\sqrt{4 - t} + 2\sqrt{2})}{(t + \frac{1}{t})} ]

[ = \lim_{t \to 0} (\sqrt{4 - t} - 2\sqrt{2})(\sqrt{4 - t} + 2\sqrt{2}) ]

  1. Apply L'Hôpital's Rule to the indeterminate form:

[ = \lim_{t \to 0} \frac{d}{dt}((\sqrt{4 - t} - 2\sqrt{2})(\sqrt{4 - t} + 2\sqrt{2})) ]

[ = \lim_{t \to 0} \frac{d}{dt}(4 - t - 8) ]

[ = \lim_{t \to 0} (-1) ]

[ = -1 ]

So, ( \lim_{t \to 0} (t + \frac{1}{t})((4 - t)^{\frac{3}{2}} - 8) = -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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