find lim x→−∞ x⋅ln(1 - 1/x).
find lim x→−∞ x⋅ln(1 - 1/x).
Answer
Explanation:
Step1: Rewrite the expression
Let (t =-\frac{1}{x}), then as (x\to-\infty), (t\to0^{+}). And (x =-\frac{1}{t}). The limit becomes (\lim_{t\to0^{+}}\left(-\frac{1}{t}\right)\ln(1 + t)).
Step2: Use the well - known limit
We know that (\lim_{t\to0}\frac{\ln(1 + t)}{t}=1). So (\lim_{t\to0^{+}}\left(-\frac{1}{t}\right)\ln(1 + t)=- 1).
Answer:
(-1)