The value of limx→∞ x3⋅sin⁡1x+x+1x2+x+1 is - Infinity Learn by Sri Chaitanya

Solution:

we have, $lim_{y \to \infty} \frac{x^{3} \cdot \sin (\frac{1}{x}) + x + 1}{x^{2} + x + 1} Put \frac{1}{x} = y$

$\begin{aligned} \Rightarrow lim_{y \to 0} \frac{\frac{1}{y^{3}} \cdot \sin (y) + \frac{1}{y} + 1}{\frac{1}{y^{2}} + \frac{1}{y} + 1} = lim_{x \to 0} \frac{\frac{1}{y^{2}} [\frac{\sin y}{y} + y + y^{2}]}{\frac{1}{y^{2}} [1 + y + y^{2}]} \\ = lim_{y \to 0} \frac{\frac{\sin y}{y} + y + y^{2}}{1 + y + y^{2}} = 1 \end{aligned}$

The value of $lim_{x \to \infty} \frac{x^{3} \cdot \sin (\frac{1}{x}) + x + 1}{x^{2} + x + 1}$ is

Solution:

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