Find the limit (if it exists). (If an answer does not exist, enter DNE.)lim Δt→0 (t + Δt)2 − 5(t + Δt) + 2 − (t2 − 5t + 2)Δt

Question

Find the limit (if it exists). (If an answer does not exist, enter DNE.)

$\lim_{{\Delta t \to 0}} \frac{{(t + \Delta t)^2 - 5(t + \Delta t) + 2 - (t^2 - 5t + 2)}}{\Delta t}$

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Solution

First, let's simplify the expression inside the limit:

(t + Δt)² - 5(t + Δt) + 2 - (t² - 5t + 2) can be expanded and simplified to:

t² + 2tΔt + Δt² - 5t - 5Δt + 2 - t² + 5t - 2

This simplifies to:

2tΔt + Δt² - 5Δt

We can factor out Δt from each term:

Δt(2t + Δt - 5)

Now, we can apply the limit:

lim Δt→0 Δt(2t + Δt - 5)

Since Δt is approaching 0, the term Δt in the expression will become 0, and we are left with:

0 * (2t - 5) = 0

So, the limit of the given expression as Δt approaches 0 is 0.

This problem has been solved

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