find the limit.\n$$\\lim_{h \\to 0} \\frac{\\sqrt{19h + 1} - 1}{h}$$\nselect the correct choice below and…

find the limit.\n$$\\lim_{h \\to 0} \\frac{\\sqrt{19h + 1} - 1}{h}$$\nselect the correct choice below and, if necessary, fill in the answer box to complete your \noa.\n$$\\lim_{h \\to 0} \\frac{\\sqrt{19h + 1} - 1}{h}=$$\n(type an integer or a simplified fraction.)\nob. the limit does not exist.

find the limit.\n$$\\lim_{h \\to 0} \\frac{\\sqrt{19h + 1} - 1}{h}$$\nselect the correct choice below and, if necessary, fill in the answer box to complete your \noa.\n$$\\lim_{h \\to 0} \\frac{\\sqrt{19h + 1} - 1}{h}=$$\n(type an integer or a simplified fraction.)\nob. the limit does not exist.

Answer

Explanation:

Step1: Rationalize the numerator

Multiply the numerator and denominator by $\sqrt{19h + 1}+1$. $$\lim_{h\rightarrow0}\frac{\sqrt{19h + 1}-1}{h}\times\frac{\sqrt{19h + 1}+1}{\sqrt{19h + 1}+1}$$ Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $(19h + 1)-1$. So the expression is $\lim_{h\rightarrow0}\frac{19h+1 - 1}{h(\sqrt{19h + 1}+1)}$.

Step2: Simplify the expression

Simplify the numerator: $19h+1 - 1 = 19h$. The expression is now $\lim_{h\rightarrow0}\frac{19h}{h(\sqrt{19h + 1}+1)}$. Cancel out the common factor $h$ (since $h\neq0$ as $h\rightarrow0$ but $h\neq0$ in the limit - taking process), we get $\lim_{h\rightarrow0}\frac{19}{\sqrt{19h + 1}+1}$.

Step3: Evaluate the limit

Substitute $h = 0$ into the expression $\frac{19}{\sqrt{19h + 1}+1}$. When $h = 0$, $\sqrt{19\times0+1}+1=\sqrt{1}+1=2$. So the limit is $\frac{19}{2}$.

Answer:

A. $\lim_{h\rightarrow0}\frac{\sqrt{19h + 1}-1}{h}=\frac{19}{2}$