Question

Evaluate the following limit:

limt→1−t2−|t−1|−1|t−1|.

Answer 1

$$\lim\limits_{t\to1}(-{t^2-|t-1|-1\over |t-1|})$$

$$\lim\limits_{t\to1^+}(-{t^2-|t-1|-1\over |t-1|})=\lim\limits_{t\to1^+}(-{t^2-t+1-1\over t-1})=$$

$$=\lim\limits_{t\to1^+}(-{t(t-1)\over t-1})=-\lim\limits_{t\to1^+}t=-1$$

$$\lim\limits_{t\to1^-}(-{t^2-|t-1|-1\over |t-1|})=\lim\limits_{t\to1^-}({t^2+t-1-1\over t-1})=$$

$$=\lim\limits_{t\to1^-}{(t+2)(t-1)\over t-1}=\lim\limits_{t\to1^-}(t+2)=3$$

$$\lim\limits_{t\to1^+}(-{t^2-|t-1|-1\over |t-1|})=-1\not=3=\lim\limits_{t\to1^-}(-{t^2-|t-1|-1\over |t-1|})$$

Therefore

$$\lim\limits_{t\to1}(-{t^2-|t-1|-1\over |t-1|})$$

does not exist.