Question

find the limit.

\\( \lim _ { h \
ightarrow 0 } \frac { \sqrt { 14 h + 1 } - 1 } { h } \\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\\( \bigcirc \\) a.
\\( \lim _ { h \
ightarrow 0 } \frac { \sqrt { 14 h + 1 } - 1 } { h } = \\)
(type an integer or a simplified fraction.)

\\( \bigcirc \\) b. the limit does not exist.

Explanation:

Step1: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator \(\sqrt{14h + 1}+1\).

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Using the formula \((a - b)(a + b)=a^{2}-b^{2}\), we get \(\lim_{h
ightarrow0}\frac{(14h + 1)-1}{h(\sqrt{14h + 1}+1)}\)

Step2: Simplify the expression

Simplify the numerator \((14h + 1)-1 = 14h\). Then the limit becomes \(\lim_{h
ightarrow0}\frac{14h}{h(\sqrt{14h+1}+1)}\). Cancel out the non - zero factor \(h\) (since \(h
ightarrow0\) but \(h
eq0\) when taking the limit), we have \(\lim_{h
ightarrow0}\frac{14}{\sqrt{14h + 1}+1}\)

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. \(\lim_{h
ightarrow0}\frac{\sqrt{14h + 1}-1}{h}=7\)