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QuestionFind the limit of the fish length model L(t)=LT−(LT−L0)e−ktL(t)=L_{T}-\left(L_{T}-L_{0}\right) e^{-k t} as t→∞t \to \infty.

Studdy Solution

STEP 1

Assumptions1. The Bertalanffy growth function is given by (t)=−(L−0)e−kt(t)=_{}-\left(L_{}-_{0}\right) e^{-k t}. . $$ is a number approaching $\infty$.
3. $k$ is a positive constant.
4. $0$ is the well-defined length of the fish at birth (when $t=0$).

STEP 2

We are asked to find lim⁡t→∞L(t)\lim{t \rightarrow \infty} L(t). This means we want to find the value of (t)(t) as tt approaches infinity.

STEP 3

Substitute tt with ∞\infty in the given function.
lim⁡t→∞L(t)=L−(L−0)e−k∞\lim{t \rightarrow \infty} L(t) = L_{}-\left(L_{}-_{0}\right) e^{-k \infty}

STEP 4

The exponential function e−k∞e^{-k \infty} approaches0 as tt approaches infinity because kk is a positive constant. Therefore, we can simplify the equation as followslim⁡t→∞L(t)=L−(L−0)⋅0\lim{t \rightarrow \infty} L(t) = L_{}-\left(L_{}-_{0}\right) \cdot0

STEP 5

Calculate the limit.
lim⁡t→∞L(t)=L−0=L\lim{t \rightarrow \infty} L(t) = L_{}-0 = L_{}So, as tt approaches infinity, the length of the fish according to the Bertalanffy growth function approaches _{}.

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