find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not\n\\( \\lim _…

Explanation:

Step1: Rationalize the expression

Multiply and divide by the conjugate: $$\lim_{x\rightarrow-\infty}\frac{(\sqrt{49x^{2}+8x}+7x)(\sqrt{49x^{2}+8x}-7x)}{\sqrt{49x^{2}+8x}-7x}$$ Using the difference of squares ((a + b)(a - b)=a^{2}-b^{2}), we get: $$\lim_{x\rightarrow-\infty}\frac{49x^{2}+8x-(49x^{2})}{\sqrt{49x^{2}+8x}-7x}=\lim_{x\rightarrow-\infty}\frac{8x}{\sqrt{49x^{2}+8x}-7x}$$

Step2: Simplify the denominator

Since (x\rightarrow-\infty), (\sqrt{49x^{2}}=-7x) (because (x<0)). Factor out (x^{2}) from the square - root: $$\lim_{x\rightarrow-\infty}\frac{8x}{|x|\sqrt{49+\frac{8}{x}}-7x}=\lim_{x\rightarrow-\infty}\frac{8x}{-x\sqrt{49+\frac{8}{x}}-7x}$$ Factor out (x) from the denominator: $$\lim_{x\rightarrow-\infty}\frac{8x}{x(-\sqrt{49+\frac{8}{x}}-7)}$$ Cancel out (x) (since (x\neq0) as (x\rightarrow-\infty)): $$\lim_{x\rightarrow-\infty}\frac{8}{-\sqrt{49+\frac{8}{x}}-7}$$

Step3: Evaluate the limit

As (x\rightarrow-\infty), (\frac{8}{x}\rightarrow0). $$\frac{8}{-\sqrt{49 + 0}-7}=\frac{8}{-7 - 7}=-\frac{8}{14}$$

Answer:

(-\frac{8}{14})