Table 4 or option D gives the correct behavior.
As x → -1⁻, r(x) → -∞
As x → -1⁺, r(x) → +∞
As x → 1⁻, r(x) → +∞
As x → 1⁺, r(x) → -∞
What is asymptote?
Let's test the asymptotic behavior of the asymptote at x = -1 and x = 1.
Asymptotic behavior describes the long-term trend of a function as its input approaches infinity or a specific value. It helps analyze how a function behaves in the limit.
From the curve
At x = -1
From left as the curve approaches the asymptote at x = -1, r(x) goes downward negative infinity.
As x → -1⁻, r(x) → -∞
From right as curve approaches the asymptote at x = -1, the curve goes upwards.
As x → -1⁺, r(x) → +∞
At x = 1
From left as the curve approaches the asymptote at x = 1, r(x) goes upwards positive infinity.
As x → 1⁻, r(x) → +∞
From right as curve approaches the asymptote at x = 1, the curve goes downwards(negative infinity)
As x → 1⁺, r(x) → -∞
Therefore, table 4 or option D gives the correct behavior.
As x → -1⁻, r(x) → -∞
As x → -1⁺, r(x) → +∞
As x → 1⁻, r(x) → +∞
As x → 1⁺, r(x) → -∞
Complete question