Problem Set - Limits with Exponential and Logarithm Functions

## Problem Set - Limits with Exponential and Logarithm Functions

 Evaluate $\underset{x\to 0}{lim}\frac{{e}^{x}-1}{x}$. Evaluate $\underset{x\to 0}{lim}\frac{{a}^{x}-1}{x}$. Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{ln}\left(1+x\right)}{x}$. Evaluate $\underset{x\to 0}{lim}{\left(1+x\right)}^{1/x}$. Evaluate $\underset{x\to 0}{lim}\frac{{a}^{x}-{b}^{x}}{x}$. Evaluate $\underset{x\to 0}{lim}\frac{{e}^{x}+{e}^{-x}-2}{{x}^{2}}$. Evaluate $\underset{x\to -\infty }{lim}{2}^{x}$. Explain why $\underset{x\to -1}{lim}\mathrm{ln}\left(x\right)$ does not exist. Explain why $\underset{x\to 0}{lim}{2}^{1/x}$ does not exist. Explain why $\underset{x\to 1}{lim}{2}^{1/\left(x-1\right)}$ does not exist. Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}$ where $f\left(x\right)={e}^{\sqrt{x}}$. Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}$ where $f\left(x\right)=ln\left(ax+b\right)$. Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}$ where $f\left(x\right)={x}^{x}$.