Questions: Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form lim h → 0 (f(x+h)-f(x))/h occur frequently in calculus. Evaluate this limit for the given value of x and function f. f(x)=√x, x=2

Solution Steps

To evaluate the limit \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) for the given function \(f(x) = \sqrt{x}\) at \(x = 2\), we can substitute \(f(x)\) into the limit expression and simplify. This limit represents the derivative of \(f(x)\) at \(x = 2\).

1. Substitute \(f(x) = \sqrt{x}\) and \(x = 2\) into the limit expression.
2. Simplify the expression inside the limit.
3. Evaluate the limit as \(h\) approaches 0.

We start with the limit expression for the derivative of the function \( f(x) = \sqrt{x} \) at \( x = 2 \): \[ \lim_{h \rightarrow 0} \frac{f(2+h) - f(2)}{h} \] Substituting \( f(x) \): \[ \lim_{h \rightarrow 0} \frac{\sqrt{2+h} - \sqrt{2}}{h} \]

The limit expression simplifies to: \[ \frac{\sqrt{2+h} - \sqrt{2}}{h} \]

To evaluate the limit as \( h \) approaches 0, we find: \[ \lim_{h \rightarrow 0} \frac{\sqrt{2+h} - \sqrt{2}}{h} = \frac{\sqrt{2}}{4} \]

Final Answer