evaluate the limit (if it exists): limit as x approaches -1 of |t+1|/(t+1)

Answer 1

When t < -1, t + 1 < 0. So, l t + 1 l = -(t + 1)

When t > -1, t + 1 > 0. So, l t + 1 l = t + 1

lim_t→--1^- [ l t + 1 l / (t + 1)] = lim_t→-1^- [-(t + 1) / (t + 1)] = -1

lim_t→-1⁺ [l t + 1 l / (t + 1] = lim_t→-1⁺ [ (t + 1) / (t + 1)] = 1

Since the one-sided limits are unequal, the limit as x→-1 does not exist.

Answer 2

I assume you mean the limit as t approaches -1

Start with t = 0. The expression evaluates to 1. As t goes negative, |t + 1| and (t + 1) both approach 0 in the same way, and the limit appears to be +1

But--start with t = -5. Now the expression evaluates to 4/-4 = -1

Thus the plot has a discontinuity at t = -1. I am tempted to say the limit is 0, but I am not sure about this