the limit represents the derivative of some function f at some number a. state such an f and a.\n\\( \\lim _…

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

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

Answer

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\rightarrow a}\frac{f(t)-f(a)}{t - a}). We want to rewrite (\lim_{t\rightarrow1}\frac{t^{4}+t - 2}{t - 1}) in the form (\lim_{t\rightarrow 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)