MCQ
$\mathop {\lim }\limits_{n \to \infty } {({3^n} + {4^n})^{\frac{1}{n}}} = $
- A$3$
- ✓$4$
- C$\infty $
- D$e$
$ = \mathop {{\rm{lim}}}\limits_{n \to \infty } \,{({4^n})^{\frac{1}{n}}}{\left[ {\frac{{{3^n}}}{{{4^n}}} + 1} \right]^{\frac{1}{n}}}$
$ = \mathop {{\rm{lim}}}\limits_{n \to \infty } 4\,{\left[ {1 + \frac{1}{{{{\left( {\frac{4}{3}} \right)}^n}}}} \right]^{1/n}}$
$ = 4\mathop {{\rm{lim}}}\limits_{n \to \infty } \,{\left[ {1 + \frac{1}{{{{\left( {\frac{4}{3}} \right)}^n}}}} \right]^{1/n}}$
$ = 4{\left[ {1 + \frac{1}{\infty }} \right]^0} = 4 \times {(1)^0}$ $ = 4 \times 1 = 4$.
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$f( x )=\frac{ x ^2-3 x -6}{ x ^2+2 x +4} \text {. }$
Then which of the following statements is (are) $TRUE$ ?
$(A)$ $f$ is decreasing in the interval $(-2,-1)$
$(B)$ $f$ is increasing in the interval $(1,2)$
$(C)$ $f$ is onto
$(D)$ Range of $f$ is $\left[-\frac{3}{2}, 2\right]$