Question

Find the limit. limₜ→₀ (e⁶ᵗ i + t²sin²t j + tan 4t k)

Answer 1

The limit of the vector function limₜ→₀ (e⁶ᵗ i + t²sin²t j + tan 4t k) is the vector i, as each component of the vector function approaches 1, 0, and 0 respectively.

Step-by-step explanation:

The student has asked to find the limit of a vector function as t approaches 0. The function is given as limₜ→₀ (e⁶ᵗ i + t²sin²t j + tan 4t k). To find this limit, we evaluate each component of the vector separately as follows:

1. For the i component: limₜ→₀ e⁶ᵗ = e⁰ = 1, since e to the power of anything that approaches 0 is 1.
2. For the j component: limₜ→₀ t²sin²t = 0, because t² approaches 0 as t approaches 0, and sin²t is bounded between 0 and 1.
3. For the k component: limₜ→₀ tan 4t = 0, since the tangent of a small angle is approximately equal to the angle itself in radians, and 4t approaches 0 as t approaches 0.

Therefore, the limit of the vector function as t approaches 0 is the vector 1 i + 0 j + 0 k, or simply i.