find the limit.\n\n\\( \\lim _ { h \\rightarrow 0 } \\frac { \\sqrt { 14 h + 1 } - 1 } { h } \\)\n\nselect…

find the limit.\n\n\\( \\lim _ { h \\rightarrow 0 } \\frac { \\sqrt { 14 h + 1 } - 1 } { h } \\)\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\n\\( \\bigcirc \\) a.\n\\( \\lim _ { h \\rightarrow 0 } \\frac { \\sqrt { 14 h + 1 } - 1 } { h } = \\)\n(type an integer or a simplified fraction.)\n\n\\( \\bigcirc \\) b. the limit does not exist.

find the limit.\n\n\\( \\lim _ { h \\rightarrow 0 } \\frac { \\sqrt { 14 h + 1 } - 1 } { h } \\)\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\n\\( \\bigcirc \\) a.\n\\( \\lim _ { h \\rightarrow 0 } \\frac { \\sqrt { 14 h + 1 } - 1 } { h } = \\)\n(type an integer or a simplified fraction.)\n\n\\( \\bigcirc \\) b. the limit does not exist.

Answer

Explanation:

Step1: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator (\sqrt{14h + 1}+1). [ \begin{align*} \lim_{h\rightarrow0}\frac{\sqrt{14h + 1}-1}{h}&=\lim_{h\rightarrow0}\frac{(\sqrt{14h + 1}-1)(\sqrt{14h + 1}+1)}{h(\sqrt{14h + 1}+1)}\ \end{align*} ] Using the formula ((a - b)(a + b)=a^{2}-b^{2}), we get (\lim_{h\rightarrow0}\frac{(14h + 1)-1}{h(\sqrt{14h + 1}+1)})

Step2: Simplify the expression

Simplify the numerator ((14h + 1)-1 = 14h). Then the limit becomes (\lim_{h\rightarrow0}\frac{14h}{h(\sqrt{14h+1}+1)}). Cancel out the non - zero factor (h) (since (h\rightarrow0) but (h\neq0) when taking the limit), we have (\lim_{h\rightarrow0}\frac{14}{\sqrt{14h + 1}+1})

Step3: Evaluate the limit

Substitute (h = 0) into (\frac{14}{\sqrt{14h+1}+1}). When (h = 0), (\sqrt{14\times0 + 1}+1=\sqrt{1}+1=2). So (\frac{14}{\sqrt{14\times0+1}+1}=\frac{14}{2}=7)

Answer:

A. (\lim_{h\rightarrow0}\frac{\sqrt{14h + 1}-1}{h}=7)