Lim t->0 (sin3tcot5t)/(tcot4t)

• odmart01
In summary, the limit as t approaches 0 of (sin3tcot5t)/(tcot4t) is equal to 12/5. This is because the cosine terms in the expression cancel out, leaving only the sine terms. By using the fact that lim theta-->0 sin(theta)/(theta) = 1, we can simplify the expression to 3/5 times 4/4, which is equal to 12/5. Therefore, the limit is 12/5.
odmart01
lim t-->0 (sin3tcot5t)/(tcot4t)

Homework Statement

lim t-->0 (sin3tcot5t)/(tcot4t)
I need to find the limit as t approaches 0.

Homework Equations

lim theta-->0 sin(theta)/(theta) =1

The Attempt at a Solution

My attempt is posted, but I'm not sure if its 0 or 12t/5

Attachments

• lop.png
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You have a "t" in the result and that can't be right! You are taking the limit as t goes to 0.

$$\frac{sin(3t)cot(5t)}{tcot(4t)}= 3\frac{sin(3t)}{3t}\frac{cos(5t)}{sin(5t)}\frac{sin(4t)}{cos(4t)}$$

The cosines are no problem, of course.
$$= 3\frac{sin(3t)}{3t}\frac{1}{5}\frac{5t}{sin(5t)}4\frac{sin(4t)}{4t}\frac{cos(5t)}{cos(4t)}$$
The two "t"s that were put into the sine fractions cancel.
Now all of those "sine" fractions go to 1 so the limit is 12/5. There is no "t".

1. What is the limit of (sin3tcot5t)/(tcot4t) as t approaches 0?

The limit of (sin3tcot5t)/(tcot4t) as t approaches 0 is not defined, as it leads to a division by 0. This type of limit is known as an indeterminate form.

2. Can L'Hôpital's rule be used to find the limit of (sin3tcot5t)/(tcot4t) as t approaches 0?

Yes, L'Hôpital's rule can be used in this case. By taking the derivatives of the numerator and denominator, the limit can be rewritten as (3cos3ttan5t - sin3t5sec2t)/(4cos4ttan4t - sin4t4sec2t). Then, as t approaches 0, the limit becomes 0/0, which can be evaluated using L'Hôpital's rule.

3. How does the value of the limit of (sin3tcot5t)/(tcot4t) change as t approaches 0 from the left and right?

The value of the limit of (sin3tcot5t)/(tcot4t) from the left and right is different. From the left, the limit approaches -∞, while from the right, it approaches ∞. This is due to the behavior of the cotangent function, which approaches -∞ from the left and ∞ from the right as t approaches 0.

4. Can the limit of (sin3tcot5t)/(tcot4t) as t approaches 0 be evaluated using a graph?

No, the limit cannot be evaluated using a graph. A graph can only provide an estimate of the limit, but cannot give a precise value. To find the exact limit, mathematical techniques such as L'Hôpital's rule or series expansion must be used.

5. Are there any restrictions on the values of t for which (sin3tcot5t)/(tcot4t) is defined?

Yes, (sin3tcot5t)/(tcot4t) is defined for all values of t except for t = 0, where the denominator becomes 0. Additionally, the cotangent functions in the numerator and denominator are undefined at multiples of π, so t cannot be equal to any of these values either.

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