Question

the limit represents the derivative of some function f at some number a. state such an f and a.
\\( \lim _ { t \
ightarrow 1 } \frac { t ^ { 4 } + t - 2 } { t - 1 } \\)
\\( f ( t ) = t - 2, a = 1 \\)
\\( f ( t ) = t ^ { 4 }, a = 1 \\)
\\( f ( t ) = t ^ { 4 } + t, a = - 1 \\)
\\( f ( t ) = t - 2, a = - 1 \\)
\\( f ( t ) = t ^ { 4 } + t, a = 1 \\)

Explanation:

Step1: Recall the definition of the derivative

The derivative of a function \(f(t)\) at \(a\) is given by \(f^{\prime}(a)=\lim_{t
ightarrow a}\frac{f(t)-f(a)}{t - a}\).
We want to rewrite \(\lim_{t
ightarrow1}\frac{t^{4}+t - 2}{t - 1}\) in the form \(\lim_{t
ightarrow a}\frac{f(t)-f(a)}{t - a}\).
Let's check the value of the function at \(t = 1\). If \(f(t)=t^{4}+t\), then \(f(1)=1^{4}+1=2\).

Step2: Substitute into the derivative formula

Now, \(\frac{f(t)-f(1)}{t - 1}=\frac{(t^{4}+t)-2}{t - 1}\).

Answer:

\(\boldsymbol{f(t)=t^{4}+t,a = 1}\) (the fifth option)