Redemitz4s

2021-11-24

Each limit represents the derivative of some function f at some number a. Indicate an f and a in each instance.
$\underset{t\to 1}{lim}\frac{{t}^{4}-t-2}{t-1}$

Aretha Frazier

Expert

Step 1
Consider the limit
$\underset{t\to 1}{lim}\frac{{t}^{4}-t-2}{t-1}$
We want to represent it as a derivative of some function f at some number a, $\mathrm{ln}$ light of definition 5 we want to write our limit in the form
$\underset{t\to a}{lim}\frac{f\left(t\right)-f\left(a\right)}{t-a}$
Since $t-1$ appears in the denominator, and $t\to 1$, we are motivated to take $a=1$
Looking at the numerator of the limit in question we want $f\left(t\right)={t}^{4}+t$. Let us see if this is the right choice. By definition 5 we have
${f}^{\prime }\left(1\right)=\underset{t\to 1}{lim}\frac{f\left(t\right)-f\left(1\right)}{t-1}$
$=\underset{t\to 1}{lim}\frac{{t}^{4}+t-\left({1}^{4}+1\right)}{t-1}$
$=\underset{t\to 1}{lim}\frac{{t}^{4}+t-2}{t-1}$
Which is what we wanted to establish.
Ans:

Hiroko Cabezas

Expert

Step 1
Definition
${f}^{\prime }\left(a\right)=\underset{x\to a}{lim}\frac{f\left(x\right)-f\left(a\right)}{x-a}$
Step 2
$\underset{t\to 1}{lim}\frac{{t}^{4}+t-2}{t-1}$
$\underset{t\to 1}{lim}\frac{\left({t}^{4}+t\right)-\left({1}^{4}+1\right)}{t-1}$
$\underset{t\to 1}{lim}\frac{f\left(t\right)-f\left(1\right)}{t-1}={f}^{\prime }\left(1\right)$
Where $f\left(t\right)={t}^{4}+t$
Note that $a=1$

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