\( \begin{array}{lll}\text { (g) } \lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}} & \text { (h) } \lim _{u \rightarrow \infty} \frac{u}{\sqrt{x^{2}+1}} & \text { (i) } \lim _{x \rightarrow \infty} \sqrt{x}(\sqrt{x+3}-\sqrt{x-2}) \\ \text { (j) } \lim _{x \rightarrow \infty} \frac{3 x+|1-3 x|}{1-5 x} & \text { (k) } \lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}) & \text { (1) } \lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+4 x}}{4 x+1} \\ \text { (m) } \lim _{x \rightarrow \infty} \frac{\sqrt{3 x^{8}+7 x^{4}}+10}{x^{4}-2 x^{2}+1} & \text { (n) } \lim _{x \rightarrow \infty}\left[\sqrt{x^{2}+5 x}-\sqrt{x^{2}-x}\right] & \text { (o) } \lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{4 x^{2}+x}-2 x}\end{array} \)
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Mind Expander
When we work with limits involving infinity, it's helpful to recognize how functions behave as they grow very large. For instance, terms of lower degree compared to the highest degree in polynomials can often be ignored when evaluating limits. This can simplify calculations immensely, especially when you apply L'Hôpital's rule for indeterminate forms! One common mistake include overlooking the behavior of logarithmic functions compared to polynomial or root functions as they approach infinity. While \( \ln x \) grows without bounds, it does so much slower than, say, \( \sqrt{x} \) or any polynomial, which could lead to miscalculations. Always consider which function dominates!
