answer the following true or false: for any function f(x) that is differentiable at x = 6. lim_{h→0} (f(6 +…

answer the following true or false: for any function f(x) that is differentiable at x = 6. lim_{h→0} (f(6 + h)-f(6))/h = lim_{x→6} (f(x)-f(6))/(x - 6). true false

answer the following true or false: for any function f(x) that is differentiable at x = 6. lim_{h→0} (f(6 + h)-f(6))/h = lim_{x→6} (f(x)-f(6))/(x - 6). true false

Answer

Explanation:

Step1: Recall derivative definition

The derivative of a function $y = f(x)$ at $x=a$ is defined in two - equivalent ways. The first way is $f^{\prime}(a)=\lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}$, and the second way is $f^{\prime}(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x - a}$.

Step2: Substitute $a = 6$

When $a = 6$, the derivative of $f(x)$ at $x = 6$ can be written as $f^{\prime}(6)=\lim_{h\rightarrow0}\frac{f(6 + h)-f(6)}{h}$ and also as $f^{\prime}(6)=\lim_{x\rightarrow6}\frac{f(x)-f(6)}{x - 6}$.

Answer:

True