Limit Problem. Any help would be appreciated.

Find the limit.

limx--->1 | x-1 | divided by (x-1)

limx---->1- x^2 - | x-1| -1 all divided by | x - 1 |

lim----->0 (sin2)/4x

How do you get the answer without its being undefined? Any help would be greatly appreciated.

1) Use 1-sided limits to carefully remove the absolute value signs.

lim(x→1-) |x - 1|/(x - 1)
= lim(x→1-) -(x - 1)/(x - 1), since x - 1 < 0 for x < 1
= -1.

lim(x→1+) |x - 1|/(x - 1)
= lim(x→1+) -(x - 1)/(x - 1), since x - 1 > 0 for x > 1
= 1.

Since the one-sided limits are not equal, the two-sided limit does not exist.
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2) This is similar in spirit to #1.

lim(x→1-) (x^2 - |x - 1| - 1)/(x - 1)
= lim(x→1-) (x^2 - -(x - 1) - 1)/(x - 1), since x - 1 < 0 for x < 1
= lim(x→1-) (x^2 + x - 2)/(x - 1)
= lim(x→1-) (x + 2)(x - 1)/(x - 1)
= lim(x→1-) (x + 2)
= 3.
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3) Assuming that you mean lim(x→0) sin(2x)/(4x):
lim(x→0) sin(2x)/(4x)
= (1/2) * lim(x→0) sin(2x)/(2x)
= 1/2 * 1, using lim(t→0) sin(t)/t = 1 with t = 2x
= 1/2.

I hope this helps!