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

Answer 1

To find the limit of (1/t - 1)/(t^2 - 2t + 1) as t approaches 1 from the right using L'Hôpital's Rule, follow these steps:

  1. Substitute 1 into the expression to obtain (1/1 - 1)/(1^2 - 2(1) + 1).
  2. Simplify the expression to get (0)/(0).
  3. Since both the numerator and denominator approach 0 as t approaches 1 from the right, apply L'Hôpital's Rule.
  4. Differentiate the numerator and denominator separately with respect to t.
  5. After differentiation, reevaluate the limit.
  6. If the new expression still approaches 0/0, repeat the process until it no longer does.
  7. The resulting expression after applying L'Hôpital's Rule will be the limit of the original expression as t approaches 1 from the right.
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Answer 2

Te limit does not exist

We seek:

# L = lim_(t rarr 1^+) (1/t-1)/(t^2-2t+1)#

From the graph it would appear that the limit does not exist:

graph{(1/x-1)/(x^2-2x+1) [-5, 5, -6, 6]}

As this is of an indeterminate form #0/0# we can apply L'Hôpital's rule.
# L = lim_(t rarr 1^+) (d/dt(1/t-1) )/( d/dt (t^2-2t+1) )#
# \ \ = lim_(t rarr 1^+) (-1/t^2 )/( 2t-2 )#
# \ \ rarr oo#, as predicted
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Answer 3

To find the limit of (1/t - 1) / (t^2 - 2t + 1) as t approaches 1 from the right using l'Hôpital's Rule, we first check if the limit evaluates to an indeterminate form, which is 0/0 or ∞/∞. If it does, we differentiate the numerator and denominator separately with respect to t, then take the limit of the resulting expression as t approaches 1 from the right.

For the numerator: lim (1/t - 1) as t approaches 1 from the right = (1/1 - 1) = 0

For the denominator: lim (t^2 - 2t + 1) as t approaches 1 from the right = (1^2 - 2(1) + 1) = 1 - 2 + 1 = 0

Since both the numerator and denominator evaluate to 0, we can apply l'Hôpital's Rule. Taking the derivatives of the numerator and denominator: Numerator: d/dt(1/t - 1) = -1/t^2 Denominator: d/dt(t^2 - 2t + 1) = 2t - 2

Now, we find the limit of the new expression as t approaches 1 from the right: lim (-1/t^2) / (2t - 2) as t approaches 1 from the right

Substitute t = 1: = (-1/1^2) / (2(1) - 2) = -1/0

Since the limit evaluates to -1/0, which is ∞, the original limit is also ∞ as t approaches 1 from the right.

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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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