Evaluate the limit, if it exists.

Question

 lim (6/t - 6/t^2 + t )

t→0

Answer 1

6/t - 6/(t²+t) = 6/t - 6/[t(t+1)] = [6(t+1) - 6] / [t(t+1)] = 6t / [t(t+1)] = 6/(t+1)

So, we have lim_t→0 [6 / (t+1)] = 6 / (0+1) = 6

Answer 2

If we substitute 0 for the values of t in the expression, this would lead due to an undefined expression since 6/t ​ and 6/t² are being divided by zero.The terms 6/t ​ and 6/t² are undefined at t=0, which means the expression as a whole is undefined at this point.

Lets assess the behavior of each term as t approaches 0:

The direction towards which t/6 ​approaches infinity as t approaches 0 depends on the direction from which t approaches 0:

If t approaches 0 from the positive side (t→0⁺), then 6/t approaches + ∞.

If t approaches 0 from the negative side (t→0^-), then 6/t ​ approaches −∞.

Since the left and right limit aren't equal to each other, this limit does not exist.

The term 6/t² ​also becomes infinitely large as t approaches 0, but it does so positively from both sides because squaring t always gives a positive result (or zero).

The addition of t to these terms slightly modifies the trajectory as t approaches 0, but it does not prevent the overall expression from tending towards negative infinity due to the dominant behavior of the 6/t²​ term.

Combining these behaviors, the dominant terms are those involving division by 6/t and 6/t² , which lead to expressions that approach infinity with opposing signs. Therefore, the limit does not exist because of the competing infinities from the first two terms.