Problems
-
Determine \[ \lim_{x\to 2}\frac{x^2+2x-24}{x-2}. \]
-
Determine \[ \lim_{x\to -3}\frac{x^3+27}{x+3}. \]
-
Prove that \begin{equation} \lim_{h\to 0}\,\frac{1}{\sqrt{x+h}+\sqrt{x}}=\frac{1}{2\sqrt{x}}. \end{equation}
-
Determine the limit \begin{equation} \lim_{x\to 3}\,\frac{x^2-9}{x-3}. \end{equation}
-
Prove that if \[ \lim_{x\to a}\,\left[f(x)+g(x)\right]=M\mbox{ and }\lim_{x\to a}\,f(x)=L \] then \[ \lim_{x\to a}\,g(x) \] exists and is equal to $M-L$.
-
Determine \begin{equation}\lim_{x\to 0}\,\frac{x^2(1+x)}{2x}.\end{equation}
-
Determine \begin{equation}\lim_{x\to 1}\,\frac{x}{x+1}.\end{equation}
-
Pinching Theorem of Function Limits
-
Prove that \begin{equation}\lim_{x\to a}\,\frac{1}{f(x)}=\frac{1}{L}.\end{equation}
-
Prove that function limits are unique.
-
Prove that \begin{equation}\lim_{x\to a}\,f(x)\,g(x)=LK.\end{equation}
-
Prove that \begin{equation}\lim_{x\to a}\,\sum_{i=1}^N\,f_i(x)=\sum_{i=1}^N\,L_i.\end{equation}
-
Prove that \begin{equation}\lim_{x\to a}\,K\,f(x)=KL\end{equation} where $K$ is a constant and \begin{equation}\lim_{x\to a}\,f(x)=L.\end{equation}
-
Prove that \[\lim_{x\to a}\,[f(x)+g(x)]=L+K.\]
