Evaluate the Limit

Problem

(lim_x→0)((x*(x2−1))/(x2))

Solution

1. Simplify the expression by canceling the common factor of x from the numerator and the denominator.

(x*(x2−1))/(x2)=(x2−1)/x

2. Analyze the behavior of the simplified expression as x approaches 0 by examining the numerator and denominator separately.

(lim_x→0)(x2−1)=−1

(lim_x→0)(x)=0

3. Determine the limit based on the vertical asymptote at x=0 Since the numerator approaches a non-zero constant and the denominator approaches zero, the limit does not exist. Specifically, we check the one-sided limits.

(lim_x→0)((x2−1)/x)=∞

(lim_x→0)((x2−1)/x)=−∞

4. Conclude that because the left-hand and right-hand limits are not equal (and both are infinite), the limit does not exist.

Final Answer

(lim_x→0)((x*(x2−1))/(x2))=Does Not Exist

Want more problems? Check here!