### Home > APCALC > Chapter 7 > Lesson 7.2.1 > Problem7-61

7-61.

No calculator! Given $f(x) = 3^x – 2$ and $g(x) =\frac { 1 } { 2 }\sin(x)$, evaluate each of the following limits.

1. $\lim\limits _ { x \rightarrow 1 }f ^\prime(x)$

$\frac{d}{dx}a^{x}=\ln(x)\cdot a^{x}$

1. $\lim\limits _ { x \rightarrow \pi / 4 } g ^ { \prime \prime } ( x )$

$\frac{d}{dx}\sin(x)=\cos(x); \frac{d}{dx}\cos(x)=-\sin(x)$

1. $\lim\limits _ { x \rightarrow \pi }f (g(x))$

$\lim \limits_{x\to\pi}(3^{0.5\sin(x)}-2)$

1. $\lim\limits _ { x \rightarrow \pi / 2 } f ( g ^ { \prime } ( x ) )$

Be mindful about positive and negative values.

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

This is Hana's definition of the derivative.

1. $\lim\limits _ { x \rightarrow \pi / 4 } \frac { g ( x ) - \frac { \sqrt { 2 } } { 4 } } { x - \frac { \pi } { 4 } }$

This is Ana's definition of the derivative.

$g'\left ( \frac{\pi }{4} \right )={\_\_\_\_\_}$