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

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

   Hint (a):

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

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

   Hint (b):

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

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

   Step (c):

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

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

   Hint (d):

   Be mindful about positive and negative values.

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

   Hint (e):

   This is Hana's definition of the derivative.

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

   Hint (f):

   This is Ana's definition of the derivative.

   Hint (f):

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