Problems : Limits

Problems for Indian Statistical Institute, Chennai Mathematical Institute, JEE Main and Advanced. $$$$ \[1.Evaluate: \lim_{x \to \infty} \frac{20+2\sqrt{x}+3\sqrt[3]{x}}{2+\sqrt{4x-3}+\sqrt[3]{8x-4}}\] \[2.Evaluate: \lim_{x \to \infty} \big( x \sqrt{x^2+a^2}-\sqrt{x^4+a^4}\big)\] \[3.Evaluate: \lim_{x \to \infty} x^3 \big\{ \sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2}x \big \}\] \[4.Evaluate: \lim_{x \to \infty} \sqrt{\frac{x-\cos^2 x}{x+\sin x}}\] \[5.Evaluate: \lim_{x \to \infty} [2\log(3x)-\log(x^2+1) ]\] 6. Let \( R_n =2+\sqrt{2+\sqrt{2+\dots+\sqrt{2}}}\) (n square roots signs). Then evaluate \(\lim_{n \to \infty} R_n \) $$$$ 7. If \(a_n = \bigg( 1+\frac{1}{n^2}\bigg)\bigg( 1+\frac{2^2}{n^2}\bigg)^2 \bigg( 1+\frac{3^2}{n^2}\bigg)^3 \dots \bigg( 1+\frac{n^2}{n^2}\bigg)^n \), then evaluate \[ \lim_{n \to \infty} a_n^{-\frac{1}{n^2}} \] $$$$ \[8.Evaluate \lim_{x \to \infty} \sqrt{x^2+x}-\sqrt{x^2+1}\] \[9. \lim_{x \to \frac{\pi}{2}} (\sin x)^{\tan x}\] \[10. \lim_{x \to 0} \frac{\cos x -1}{\sin^2 x}\] For PDF click here