91影视

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$is positive

$\underset{x鈫�鈭�}{\lim }{e}^{kx}=?.$

Write the difference rule for limits in terms of delta鈥揺psilon statements.

Use limit rules and the continuity of polynomial functions to prove that every rational function is continuous on its domain.

Write a delta鈥揺psilon proof that proves that $f$is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to $1$.

$f\left(x\right)=x^3$

Prove the second part of Theorem$1.31$(a): If $k>0$, then $\underset{x鈫�鈭�}{\lim }{x}^{-k}=0$.

Prove the constant multiple rule for limits:
$\underset{\mathbf{x}\mathbf{鈫�}\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{L}\mathbf{\text{and}}\mathit{k}\mathbf{鈭�}\mathbf{鈩�}\mathbf{,}\mathbf{\text{then}}\underset{\mathbf{x}\mathbf{鈫�}\mathbf{c}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{k}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{k}\mathit{L}$

Write a delta鈥揺psilon proof that proves that $f$is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to $1$.

$f\left(x\right)=x^{鈭�2}$

Prove the $k=1$ case of the first part of Theorem 1.31(b): that $\underset{x鈫�鈭�}{\lim }{e}^x=鈭�$. (Hint: Given $M>0$, choose $N=\ln M$. Then if $x>N=\ln M$, we must have $x=\ln M+c$ for some positive number c. Use this to show that $e^x>M$.)

Write a delta鈥揺psilon proof that proves that $f$is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to $1$.

$f\left(x\right)=x^{鈭�1/2}$

Prove the difference rule for limits by applying the sum and constant multiple rules for limits.