6.5. $$ \lim _{x \rightarrow +\infty} \frac{2 x+5}{x^{2}-7 x+3} $$.

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Calculate the limit $$ \lim_{x\rightarrow +\infty} \frac{2x+5}{x^{2}-7x+3} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{x\rightarrow +\infty}\left(\frac{2x+5}{x^{2}-7x+3}\right)$$
- step1: Rewrite the expression:
  $$\lim _{x\rightarrow +\infty}\left(\frac{x\left(2+\frac{5}{x}\right)}{x^{2}\left(1-\frac{7}{x}+\frac{3}{x^{2}}\right)}\right)$$
- step2: Reduce the fraction:
  $$\lim _{x\rightarrow +\infty}\left(\frac{2+\frac{5}{x}}{x\left(1-\frac{7}{x}+\frac{3}{x^{2}}\right)}\right)$$
- step3: Rewrite the expression:
  $$\frac{\lim _{x\rightarrow +\infty}\left(2+\frac{5}{x}\right)}{\lim _{x\rightarrow +\infty}\left(x\left(1-\frac{7}{x}+\frac{3}{x^{2}}\right)\right)}$$
- step4: Calculate:
  $$\frac{2}{\lim _{x\rightarrow +\infty}\left(x\left(1-\frac{7}{x}+\frac{3}{x^{2}}\right)\right)}$$
- step5: Calculate:
  $$\frac{2}{+\infty}$$
- step6: Calculate:
  $$0$$
The limit of the expression $$ \frac{2x+5}{x^{2}-7x+3} $$ as $$ x $$ approaches positive infinity is 0.
The limit is 0.

6.5. .

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