Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 7 }$.
A) $\frac { 1 } { 7 \sqrt { x - 7 } }$
B) $\frac { 1 } { 2 \sqrt { x - 7 } }$
C) $\frac { \sqrt { x - 7 } } { 2 }$
D) $\sqrt { x - 7 }$
E) limit does not exist

Question

Quizplus · Accepted Answer

We have, 
$$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } = \lim _ { h \rightarrow 0 } \frac { \sqrt { ( x + h ) -7 } - \sqrt { x -7 } } { h }$$
Now, we multiply and divide numerator by $\sqrt { ( x + h ) -7 } + \sqrt { x -7 }$. Thus, we have,
$$\lim _ { h \rightarrow 0 } \frac { \sqrt { ( x + h ) -7 } - \sqrt { x -7 } } { h } = \lim _ { h \rightarrow 0 } \frac { ( x + h ) - 7 - x + 7 } { h \left( \sqrt { ( x + h ) -7 } + \sqrt { x -7 }\right) } = \lim _ { h \rightarrow 0 } \frac { 1 } {\left( \sqrt { ( x + h ) -7 } + \sqrt { x -7 }\right) } = \frac { 1 } { 2 \sqrt { x - 7 } }$$
Thus, option B is the correct choice.

Find 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)​

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$