Evaluate the following limits, show all your work with good form and indicate which algebraic tools you used. (substitution, factoring, rationalizing, change of variable)
1. $\lim _{x_{-} \rightarrow 3} \frac{x^{3}-21}{2 x+3}$
2. $\lim _{x_{-} \rightarrow 3} \frac{9-x^{2}}{3-x}$
3. $\lim _{x_{-} \rightarrow 0} \frac{\sqrt{1+4 x}-\sqrt{1+x}}{x}$
4. $\lim _{h \rightarrow 0} \frac{(1+h)^{\frac{1}{6}}-1}{h}$
Answer
Final Answer: The limit of the function \(\frac{x^{3}-21}{2 x+3}\) as \(x\) approaches \(3\) from the left is \(\boxed{\frac{2}{3}}\).
Step 1 ：The first question is asking for the limit of the function \(\frac{x^{3}-21}{2 x+3}\) as \(x\) approaches \(3\) from the left.
Step 2 ：To solve this, we can simply substitute \(x=3\) into the function and evaluate it.
Step 3 ：This is because the function is continuous at \(x=3\), and the limit from the left is equal to the function value at that point.
Step 4 ：Substituting \(x=3\) into the function, we get \(\frac{3^{3}-21}{2 \cdot 3+3} = \frac{2}{3}\).
Step 5 ：Final Answer: The limit of the function \(\frac{x^{3}-21}{2 x+3}\) as \(x\) approaches \(3\) from the left is \(\boxed{\frac{2}{3}}\).
