Questions: Evaluate each of the following limits: xi) lim x→2 (x-2)/(x-2) xii) lim x→3/2 (2x^2-3x)/2x-3.

Transcript text: Evaluate each of the following limits: xi) $\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}$ xii) $\lim _{x \rightarrow \frac{3}{2}} \frac{2 x^{2}-3 x}{|2 x-3|}$.

Solution

Solution Steps

Step 1: Evaluate limit xi)

We need to evaluate the limit:

lim (|x-2|)/(x-2)
x->2

We consider the left-hand and right-hand limits separately.

Left-hand limit: When x approaches 2 from the left (x < 2), |x-2| = -(x-2). Therefore,

lim (|x-2|)/(x-2) = lim -(x-2)/(x-2) = lim -1 = -1
x->2-            x->2-            x->2-

Right-hand limit: When x approaches 2 from the right (x > 2), |x-2| = x-2. Therefore,

lim (|x-2|)/(x-2) = lim (x-2)/(x-2) = lim 1 = 1
x->2+            x->2+            x->2+

Since the left-hand limit (-1) and the right-hand limit (1) are not equal, the limit does not exist.

Step 2: Evaluate limit xii)

We need to evaluate the limit:

lim |(2x^2 - 3x)| / |2x-3|
x->3/2

We can rewrite the expression as:

lim |x(2x-3)| / |2x-3|
x->3/2

Since x approaches 3/2, x is close to 3/2 but not equal to 3/2. Therefore, 2x-3 is not equal to 0 and we can simplify the expression:

lim |x(2x-3)| / |2x-3| = lim |x| = |3/2| = 3/2
x->3/2                  x->3/2

Final Answer:

The limit xi) does not exist.

The limit xii) is 3/2.

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