# Find Limit of Vector: i+(Sin(4t))/(4t)+e^(-5t)k

• iberhammer
In summary, the limit for the given function is undefined, or DNE. The approach of plugging in 0 did not work, and using a calculator to find a decimal answer also did not work. This is because the j coordinate has a limit that is related to a well-known limit from calculus, which is equal to 1. Therefore, the limit of the given function does not exist.
iberhammer
Find the limit (if it exists). (If an answer does not exist, enter DNE.)
lim t → 0
e^9t i + (Sin(4t))/(4t) j + e^(-5t) k

My approach was to plug in 0 and get i+k but that didn't go through with webassign. So I tried the DNE answer because maybe since the y could be undefined, the whole thing is, but that was wrong. So lastely I punched it into my calculator, got the decimal answer and that didn't work for i + (decimal) j + k.

Does anyone know how to solve for the limit?

The only tricky one here is the limit for the j coordinate,
$$\lim_{t \to 0} \frac{sin(4t)}{4t}$$

This is related to the following well-known limit from calculus,
$$\lim_{t \to 0} \frac{sin(t)}{t} = 1$$

## 1. What is the limit of the given vector?

The limit of the given vector can be found by taking the limit of each component separately. The limit of the x-component (i) is 1, the limit of the y-component (Sin(4t))/(4t) is 1, and the limit of the z-component e^(-5t)k is 0. Therefore, the limit of the vector is (1, 1, 0).

## 2. How do you find the limit of a vector with trigonometric functions?

To find the limit of a vector with trigonometric functions, you can use the fact that the limit of a product is equal to the product of the limits. In this case, you can take the limit of each component separately and then combine them to find the limit of the vector.

## 3. Is the limit of a vector always a point?

No, the limit of a vector can also be a line or a plane in certain cases. This happens when the components of the vector approach different values as the independent variable approaches a certain value.

## 4. Can the limit of a vector be infinity?

Yes, the limit of a vector can be infinity if one or more of its components approaches infinity as the independent variable approaches a certain value. In such cases, the direction of the vector will also be important in determining the limit.

## 5. How do you determine if the limit of a vector exists?

The limit of a vector exists if and only if the limit of each component of the vector exists. This means that all the components must approach a finite value as the independent variable approaches a certain value. If even one component has a limit of infinity or does not exist, then the limit of the vector does not exist.

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