Question

What is the value of the following limit as x approaches 0?

lim (1 + x⁻¹ + x³ - 1 + x⁴ - 1 + x⁵) / lim (4/(1+x) - 5/(1+x) + 1/(1+x) - 3/(1+x))

A) 1

B) 0

C) Undefined

Answer 1

The limit of the expression as x approaches 0 is undefined because the limit of the numerator goes to infinity, whereas the limit of the denominator is a finite number (-3). Thus, we have an undefined value due to division by zero.

Step-by-step explanation:

To find the value of the limit as x approaches 0 for the expression given, we should first simplify the expression in the numerator and denominator before taking the limit. The numerator simplifies to x⁻¹ + x³ + x⁴ + x⁵, and the denominator simplifies to (4 - 5 + 1 - 3)/(1+x), which simplifies to -3/(1+x).

Now, let's take the limits separately for the numerator and denominator:

• The limit of the numerator as x approaches 0 is undefined because x⁻¹ (1/x) goes to infinity as x approaches 0.
• The limit of the denominator as x approaches 0 is -3, as 1+x approaches 1, making the fraction -3/1.

As the numerator's limit is undefined and the denominator's limit is a finite number, the overall limit is undefined.