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Recent questions in Limits and continuity
Limits and continuity

### Evaluate the following limit using the substitution $$\displaystyle{t}={x}^{{{6}}}$$ $$\displaystyle\lim_{{{t}\rightarrow{1}}}{b}{i}{g}{g{{\left({\frac{{\sqrt{{{3}}}{\left\lbrace{t}\right\rbrace}-{1}}}{{\sqrt{{{t}}}-{1}}}}{b}{i}{g}{g}\right)}}}$$

Limits and continuity

### Find the following limits 1) $$\displaystyle\lim_{{{x}\to{0}}}\frac{{\tan{{\left({4}{x}\right)}}}}{{{x}+{\sin{{\left({2}{x}\right)}}}}}$$ 2) $$\displaystyle\lim_{{{x}\to\infty}}{\left({e}^{{{2}{x}}}+{x}\right)}^{{\frac{{1}}{{x}}}}$$

Limits and continuity

### Evaluate the limits, using algebra and/or limit properties as needed. $$\displaystyle\lim_{{{x}\rightarrow{2}}}{\frac{{{3}{x}^{{2}}-{5}}}{{{x}^{{2}}+{3}{x}-{3}}}}$$

Limits and continuity

### Find the limit (if it exists) and discuss the continuity of the function. $$\displaystyle\lim_{{{x},{y}}}\rightarrow{\left({1},{1}\right)}{\frac{{{x}{y}}}{{{x}²+{y}²}}}$$

Limits and continuity

### Find the limit and discuss the continuity of the function. $$\displaystyle\lim_{{{{x},{y}}}\rightarrow{\left({\frac{{\pi}}{{{4}}}},{2}\right)}}{y}{\cos{{x}}}{y}$$

Limits and continuity

### Use the limit definition of the derivative to calculate the derivatives of the following function: (a) $$f(x)=4x^2+3x+1$$ (b) $$f(x)=\frac{2}{x^2}$$

Limits and continuity

### Find the limit and discuss the continuity of the function. $$\lim_{x,y \rightarrow 2\pi,4} \sin \frac{x}{y}$$

Limits and continuity

### Find the limit and discuss the continuity of the function. $$\displaystyle\lim_{{{x},{y}}}\rightarrow{\left(-{1},{2}\right)}{\frac{{{\left({x}+{y}\right)}}}{{{\left({x}-{y}\right)}}}}$$

Limits and continuity

### Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. a) $$\lim \frac{2n+1}{5n+4}=\frac{2}{5}$$ b) $$\lim \frac{2n^3}{n^3+3}=0$$ c) $$\lim \frac{\sin (n^2)}{\sqrt[3]{n}}$$

Limits and continuity

### a) Using limit rules each of the following limits $$\lim_{x\rightarrow0}\frac{x+1}{x}$$ $$\lim_{x\rightarrow0}\frac{x+1}{x^{2}}$$ $$\lim_{x\rightarrow0}\frac{x}{x+1}$$ $$\lim_{x\rightarrow0}\frac{x+1}{x+2}$$ b) Using limit rules evaluate $$\lim_{x\rightarrow2}\frac{\sqrt{x+7}-3}{x-2}$$

Limits and continuity

### Find the limit and discuss the continuity of the function. $$\displaystyle\lim_{{{x},{y},{z}}}\rightarrow{\left({1},{3},{4}\right)}\sqrt{{{\left({x}+{y}+{z}\right)}}}$$

Limits and continuity

### Evaluate the limits using algebra and|or limit properties as needed

Limits and continuity

### Find the limit (if it exists) and discuss the continuity of the function. $$\lim_{(x,y,z) \rightarrow (-3,1,2)}\frac{\ln z}{xy-z}$$

Limits and continuity

### The limit as x approaches zero of $$\displaystyle\frac{{{\sin{{x}}}-{x}{\cos{{x}}}}}{{x}^{{3}}}$$

Limits and continuity