Find the following limit using algeraic manipulations. (Give excat asnwers).
lim (4t^2 -t-1)/(t^2 -1)
t--> -1
To find the limit of the given function using algebraic manipulations, we can simplify the expression by factoring both the numerator and the denominator.
Step 1: Factor the numerator and denominator.
The numerator can be factored as follows: 4t^2 - t - 1 = (t - 1)(4t + 1).
The denominator is a difference of squares: t^2 - 1 = (t - 1)(t + 1).
Step 2: Cancel out common factors.
Now that we have factored both the numerator and denominator, we can cancel out the common factor of (t - 1).
The expression simplifies to: (4t + 1)/(t + 1).
Step 3: Substitute the limit value.
To find the limit as t approaches -1, we substitute -1 into the simplified expression:
lim (4t + 1)/(t + 1) = (4(-1) + 1)/(-1 + 1) = (-4 + 1)/(0) = -3/0.
Step 4: Determine the type of the limit.
For the given limit, we have an indeterminate form of -3/0, which indicates that the limit does not exist.
Therefore, the limit of (4t^2 - t - 1)/(t^2 - 1) as t approaches -1 does not exist.