Find the limit ( e^-3t i + tsin(1/t)j + arctan tk ) if it exists. If not say doesn’t exsist.
(t→ ∞)
To find the limit of a vector function, we need to take the limit of each component separately.
Let's break down the given vector function:
F(t) = e^(-3t)i + t*sin(1/t)j + arctan(tk)
As t approaches infinity (t → ∞), we need to find the limits of each component individually.
1) For the first component, e^(-3t)i, the exponential term approaches 0 as t approaches infinity. So, the limit of this component is 0i = 0.
2) For the second component, t*sin(1/t)j, we know that sin(1/t) is bounded between -1 and 1 for all values of t. As t approaches infinity, the term t*sin(1/t) approaches 0. Therefore, the limit of this component is 0j = 0.
3) For the third component, arctan(tk), we need to consider the behavior of arctan(tk) as t approaches infinity. The arctan function is defined between -π/2 and π/2. As t gets larger, the term tk grows without bound. Since there are no restrictions on the magnitude of k, the limit of this component does not exist. Therefore, the limit of this component, and hence the entire vector function F(t), does not exist as t approaches infinity.
In conclusion, the limit of the given vector function F(t) = e^(-3t)i + t*sin(1/t)j + arctan(tk) as t approaches infinity (t → ∞) does not exist.