The limit represents the derivative of some function f at some number a. Select an appropriate f(t) and a.
lim_(t->1)(t^3+t-2)/(t-1)
1f(t) = t^3, a = 1
f(t) = t^3 + t, a = 1
f(t) = t - 2, a = 1
f(t) = t - 2, a = -1
f(t) = t^3 + t, a = -1
f(t) = t^3, a = -1
none of these
Lim (t^3 + t - 2)/(t-1) as t ---> 1
= lim((t-1)(t^2 + t + 2)/t-1)
= lim t^2 + t + 2
= 1 + 1 + 2
= 4
Don't see how the rest of your choices tie in with that.
To find an appropriate function f(t) and number a such that the given limit represents the derivative, we need to consider the derivative rules.
The given limit is lim_(t->1) (t^3 + t - 2)/(t - 1).
To represent the derivative, we look for a function f(t) which, when differentiated, will give us (t^3 + t - 2)/(t - 1).
Taking the derivative of f(t), we get f'(t) = 3t^2 + 1.
Comparing this with (t^3 + t - 2)/(t - 1), we see that they are not equal.
Therefore, none of the options provided is appropriate for f(t) and a in this case.
To find an appropriate function f(t) and number a for which the limit represents the derivative, we need to consider the concept of the derivative itself.
The derivative of a function represents the slope of the tangent line to the graph of the function at a particular point. So, for the limit to represent the derivative, we need to choose a function and a point where the limit expression actually gives us the slope of the tangent line at that point.
In this case, we have the limit expression lim_(t->1) (t^3 + t - 2)/(t - 1). To find an appropriate f(t) and a value for a, we can rewrite the expression in a way that resembles the definition of the derivative.
(t^3 + t - 2)/(t - 1) can be factored as (t - 1)(t^2 + t + 2)/(t - 1). Notice that we have (t - 1) in both the numerator and denominator. We can simplify the expression by canceling out the common factor:
(t^2 + t + 2)/(1).
Now, let's choose a function f(t) that represents this expression. We can select f(t) = t^2 + t + 2, and a value for a as a = 1.
Therefore, the appropriate f(t) and a values that make the limit represent the derivative of f at a are f(t) = t^2 + t + 2 and a = 1.