Question

In Exercises \(31 - 36 ,\) determine the limit. $$\lim _ { x \rightarrow 2 ^ { - } } \operatorname { int } x$$

Short answer

The limit of the function as x approaches 2 from the left is 1.

Step-by-step solution

Understand the Function

The function given is the integer part function (floor function). This function gives the largest integer less than or equal to x. So, for any integer value x, \(\operatorname { int } x = x\). And for any non-integer x, \(\operatorname { int } x\) is the integer part of x, which is the largest integer less than x.

Apply the Limit

The limit is given as \(x \rightarrow 2^-\), indicating that x is approaching 2 from the left or the negative side. So, the values of x are slightly less than 2, but not equal to 2. When these values are placed into the integer part function, we get integers less than 2.

Determine the Limit

As x is approaching 2 from the left side, all the numbers being considered are less than 2 but very close to 2. So the greatest integer value less than these fractions would be 1. Hence, the limit of the function as x approaches 2 from the left side is 1.