<s>[INST] Evaluating $\lim_{x\to 0} \frac{1-\sqrt{3x+1}}{2-\sqrt{5x+4}}$ $$ \lim_{x\to 0} \frac{1-\sqrt{3x+1}}{2-\sqrt{5x+4}}$$
How do I solve this without using derivatives or integrals.
 [/INST] Take a look at the following steps
$$\eqalign{
  & L = \mathop {\lim }\limits_{x \to 0} {{1 - {{(3x + 1)}^{0.5}}} \over {2 - {{(5x + 4)}^{0.5}}}} = \mathop {\lim }\limits_{x \to 0} {{1 - {{(3x + 1)}^{0.5}}} \over {2 - {{(5x + 4)}^{0.5}}}}{{2 + {{(5x + 4)}^{0.5}}} \over {2 + {{(5x + 4)}^{0.5}}}}{{1 + {{(3x + 1)}^{0.5}}} \over {1 + {{(3x + 1)}^{0.5}}}}  \cr 
  & \,\,\,\, = \mathop {\lim }\limits_{x \to 0} {{1 - (3x + 1)} \over {4 - (5x + 4)}}{{2 + {{(5x + 4)}^{0.5}}} \over {1 + {{(3x + 1)}^{0.5}}}}  \cr 
  & \,\,\,\, = \mathop {\lim }\limits_{x \to 0} {{ - 3x} \over { - 5x}}{{2 + {{(5x + 4)}^{0.5}}} \over {1 + {{(3x + 1)}^{0.5}}}}  \cr 
  & \,\,\,\, = \mathop {\lim }\limits_{x \to 0} {{ - 3x} \over { - 5x}}\mathop {\lim }\limits_{x \to 0} {{2 + {{(5x + 4)}^{0.5}}} \over {1 + {{(3x + 1)}^{0.5}}}}  \cr 
  & \,\,\,\, = \left( {{3 \over 5}} \right)\left( {{4 \over 2}} \right) = {6 \over 5} \cr} $$
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