Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ occur frequently in calculus. Evaluate this limit for the given value of $x$ and function $f$.
-$$f ( x ) = \sqrt { x } , x = 13$$
A) Does not exist
B) $\frac { \sqrt { 13 } } { 13 }$
C) $\frac { 13 } { 2 }$
D) $\frac { \sqrt { 13 } } { 26 }$

Question

Quizplus · Accepted Answer

This limit represents the derivative of $f(x) = \sqrt{x}$ at $x = 13$. The derivative of $f(x) = \sqrt{x}$ is $f'(x) = \frac{1}{2\sqrt{x}}$. Substituting $x = 13$ gives $f'(13) = \frac{1}{2\sqrt{13}} = \frac{\sqrt{13}}{26}$.

Because of Their Connection with Secant Lines, Tangents, and Instantaneous lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​

Because of Their Connection with Secant Lines, Tangents, and Instantaneous $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$