For limits of the form $\lim_{x \to \infty} (\sqrt{x^2 + bx + c} - x)$, there is a beautiful shortcut formula: the result is always exactly equal to $\frac{b}{2}$! Here, $b = 5$, so the answer is instantly $\frac{5}{2}$. This shortcut saves an incredible amount of scratchpad work!
Step 1: Identify the trouble.
As $x$ grows large, $\sqrt{x^2+5x-7}$ behaves almost like $x$, so we get $\infty-\infty$. We fix this by multiplying by the conjugate.
Step 2: Multiply by the conjugate.
Multiply top and bottom by $\sqrt{x^2+5x-7}+x$. The top becomes $(x^2+5x-7)-x^2=5x-7$. \[ \lim_{x\to\infty}\frac{5x-7}{\sqrt{x^2+5x-7}+x} \]
Step 3: Divide through by $x$.
\[ \lim_{x\to\infty}\frac{5-\frac{7}{x}}{\sqrt{1+\frac{5}{x}-\frac{7}{x^2}}+1} \]
Step 4: Let the small terms vanish.
Every term with $x$ in the bottom goes to 0. \[ \frac{5}{1+1}=\frac{5}{2} \] \[ \boxed{\frac{5}{2}} \]