Question

which limit below is equal to the slope m of the curve at the given point?
a. lim h→0 √10(10+h) - √10(10) / h
b. lim h→0 √10(10) - √10(10+h) / h
c. lim h→0 √10+h - √10 / h
d. lim h→0 √10(10 - h) + √10(10) / h

Explanation:

Step1: Recall the formula for the slope of the tangent line

The slope \(m\) of the tangent line to the curve \(y = f(x)\) at the point \((a,f(a))\) is given by \(m=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}\).

Step2: Identify \(f(x)\) and \(a\)

Here, \(f(x)=\sqrt{10x}\) and \(a = 10\). Then \(f(10)=\sqrt{10\times10}=\sqrt{100} = 10\) and \(f(10 + h)=\sqrt{10(10 + h)}\).

Step3: Substitute into the slope formula

Substituting \(f(10 + h)\) and \(f(10)\) into the formula \(m=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}\), we get \(m=\lim_{h
ightarrow0}\frac{\sqrt{10(10 + h)}-\sqrt{10\times10}}{h}\).

Answer:

A. \(\lim_{h
ightarrow0}\frac{\sqrt{10(10 + h)}-\sqrt{10\times10}}{h}\)