Question

Assertion(A): $\underset{x\to \infty }{lim}{\left(\frac{x^2+5x+3}{x^2+x+2}\right)}^x=e^4$
Reason(R): $\underset{x\to \infty }{lim}(1+x{)}^{1/x}=e$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. R is true but A is false

Answer 1

Answer: (C)

A is true but R is false

$\underset{x\to \infty }{lim}{\left(\frac{x^2+5x+3}{x^2+x+2}\right)}^x,1^{\infty }$ form.

$=\underset{x\to \infty }{lim}{\left(1+\frac{4x+1}{x^2+x+2}\right)}^x$

$Now,\underset{x\to 0}{lim}f(x{)}^{g\left(x\right)}=e^{(\underset{x\to 0}{lim}\left(f\right(x)-1).g\left(x\right))}$
Using above formula we get,

$=e^{\underset{x\to \infty }{lim}\frac{4x^2+4}{x^2+x+2}}=e^4$
Hence Assertion is true. Reason is obviously Incorrect.