9. Which of the following statement(s) is/are true? (1) \( \lim _{x \rightarrow a} \frac{a^{x}-x^{a}}{x^{x}-a^{a}}=-1 \) then \( a \) is 1 (2) \( \lim _{n \rightarrow \infty} \log _{(n-1)} n \cdot \log _{n}(n+1) \cdot \log _{(n+1)}(n+2) \ldots . . \log _{\left(n^{5}-1\right)} n^{5}=5 \) (3) \( \lim _{x \rightarrow 0} \log _{\frac{x^{2}+1}{x+1}}\left(\sqrt{x^{2}+1}-x\right)=1 \) (4) \( \lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2}}{y}=0 \), when \( (x, y) \rightarrow(0,0) \) along the curve \( y^{2}=x^{3} \)

Question

9. Which of the following statement(s) is/are true? (1) \( \lim _{x \rightarrow a} \frac{a^{x}-x^{a}}{x^{x}-a^{a}}=-1 \) then \( a \) is 1 

(2) \( \lim _{n \rightarrow \infty} \log _{(n-1)} n \cdot \log _{n}(n+1) \cdot \log _{(n+1)}(n+2) \ldots . . \log _{\left(n^{5}-1\right)} n^{5}=5 \)

 (3) \( \lim _{x \rightarrow 0} \log _{\frac{x^{2}+1}{x+1}}\left(\sqrt{x^{2}+1}-x\right)=1 \) 

(4) \( \lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2}}{y}=0 \), when \( (x, y) \rightarrow(0,0) \) along the curve \( y^{2}=x^{2} \)

Comments

Its y² = x² in last option

Answer 1

Answer(1) 

∴ ln a-1 = -1 - ln a

= 2ln a =0

∴ a = 1