Question

Find the following limit or state that it does not exist. Assume b is a fixed real number.

lim (x-b)²⁰-x+b/x-b
x-b

Answer 1

To find the limit of the given expression, simplify it and substitute x with the given value. The limit does not exist in this case.

Step-by-step explanation:

To find the limit of the given expression, we can simplify it first. So, let's simplify the expression using algebraic steps:

lim (x-b)²⁰ - (x+b) / (x-b)(x-b)

= lim [(x-b)² - (x+b)] / (x-b)²

= lim [(x² - 2bx + b²) - (x + b)] / (x-b)²

= lim (x² - x - 2bx + b² - b) / (x-b)²

= lim (x² - 2bx - x + b² - b) / (x-b)²

= lim (x² - (2bx + x) + (b² - b)) / (x-b)²

= lim x² - 3bx + (b² - b) / (x-b)²

Now, as x approaches b, we can substitute x with b in the expression:

= b² - 3b² + (b² - b) / (b-b)²

= b² - 3b² + (b² - b) / 0

Since the denominator is zero, it means that the limit does not exist.