Use the \(\epsilon\), \(\delta\) definition of a limit to prove the following statement. \[\lim_{x\rightarrow3}\frac{x}{5}=\frac{3}{5}\]

Use the \(\epsilon\), \(\delta\) definition of a limit to prove the following statement. \[\lim_{x\rightarrow2}\frac{x^2+x-6}{x-2}=5\]

Use \(\epsilon\), \(\delta\) definition of a limit to prove the following statement. \[\lim_{x\rightarrow 3}x^2=9\]

What are the limit theorems, and how do we use them?

Find the following limits.

1. \(\displaystyle\lim_{x\rightarrow2}\left(5-x^2\right)\)

2. \(\displaystyle\lim_{x\rightarrow3}\displaystyle\frac{x^2-4}{x+2}\)

3. \(\displaystyle\lim_{x\rightarrow1}\sqrt{x^2+3}\)

4. \(\displaystyle\lim_{x\rightarrow2}\displaystyle\frac{1}{2}\sin x\)

5. \(\displaystyle\lim_{x\rightarrow e}4\ln x\)

6. \(\displaystyle\lim_{x\rightarrow0}3e^x\)

Find the following limits.

1. \(\displaystyle\lim_{x\rightarrow4}\displaystyle\frac{x^2-4}{3x-6}\)

2. \(\displaystyle\lim_{x\rightarrow0}\displaystyle\frac{x^2-4}{3x-6}\)

3. \(\displaystyle\lim_{x\rightarrow2}\displaystyle\frac{x^2-4}{3x-6}\)

Evaluate: \(\displaystyle\lim_{x\rightarrow2}\displaystyle\frac{x^2+x-6}{x-2}\)

AP Calculus Exam: Multiple Choice Question

Find the following limits.

1. \(\displaystyle\lim_{\theta\rightarrow0}\displaystyle\frac{\sin\theta}{\theta}\)

2. \(\displaystyle\lim_{\theta\rightarrow0}\displaystyle\frac{\theta}{\sin\theta}\)