[tex]\boxed{\begin{aligned} \sf \lim_{x \to 2} \frac{ \sqrt{ {x}^{2} + 5 } - \sqrt{ {5x}^{2} - 4 } }{ {4x}^{2} - 9 } &= \sf \frac{ \sqrt{ {2}^{2} + 5} - \sqrt{5 {(2)}^{2} - 4 } }{4 {(2)}^{2} - 9 } \\ \\ \sf &= \sf \frac{ \sqrt{4 + 5} - \sqrt{20 - 4} }{16 - 9} \\\\ \sf &= \sf \frac{ \sqrt{9} - \sqrt{16} }{7} \\ \\ \sf &= \sf \frac{3 - 4}{7} \\ \\ \sf &= \sf - \frac{1}{7}\end{aligned}}[/tex]
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[tex]\boxed{\begin{aligned} \sf \lim_{x \to 2} \frac{ \sqrt{ {x}^{2} + 5 } - \sqrt{ {5x}^{2} - 4 } }{ {4x}^{2} - 9 } &= \sf \frac{ \sqrt{ {2}^{2} + 5} - \sqrt{5 {(2)}^{2} - 4 } }{4 {(2)}^{2} - 9 } \\ \\ \sf &= \sf \frac{ \sqrt{4 + 5} - \sqrt{20 - 4} }{16 - 9} \\\\ \sf &= \sf \frac{ \sqrt{9} - \sqrt{16} }{7} \\ \\ \sf &= \sf \frac{3 - 4}{7} \\ \\ \sf &= \sf - \frac{1}{7}\end{aligned}}[/tex]