# Recent questions in Limits and continuity

Limits and continuity

### Use Taylor series to evaluate the following limits. $$\lim_{x\rightarrow0}\frac{\sec x-\cos x-x^2}{x^4} \ (Hint: \text{The Maclaurin series for sec x is }1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+...)$$

Limits and continuity

### Find the limits. Write $$\infty$$ or $$-\infty$$ where appropriate. $$\lim_{x\rightarrow0^-}\frac{x^2-3x+2}{x^3-4x}$$

Limits and continuity

### Evaluate the following limits: $$\lim_{y\rightarrow0}\frac{1-\cos(7y)}{2y}$$

Limits and continuity

### Find the limit. $$\lim_{x\rightarrow(\pi/2)}\frac{\sec x}{\tan x}$$

Limits and continuity

### Evaluate the following limits. $$\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{yz-xy-xz-x^2}{yz+xy+xz-y^2}$$

Limits and continuity

### Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate. $$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}$$

Limits and continuity

### Find the limits: As $$x\rightarrow\infty, \frac{-2x(x^2+4)}{-7-7x^4}$$

Limits and continuity

### Suppose the functions f(x) and g(x) are defined for all x and that $$\lim_{x\rightarrow0}f(x)=\frac{1}{2}$$ and $$\lim_{x\rightarrow0}g(x)=\sqrt2$$. Find the limits as $$x\rightarrow0$$ of the following functions. $$f(x)\frac{\cos x}{x-1}$$

Limits and continuity

### Evaluate the following limit. $$\lim_{x\rightarrow0}\frac{\sin ax-\tan^{-1}ax}{bx^3}$$

Limits and continuity

### Evaluate the following limits. $$\lim_{(x,y)\rightarrow(1,2)}\frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$

Limits and continuity

### Use the method of your choice to evaluate the following limits. $$\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{2x^2-xy-y^2}$$

Limits and continuity

### Evaluate the following limits. $$\lim_{x\rightarrow2}\frac{\sin(x-2)}{x^2-4}$$

Limits and continuity

### Find the limit: $$\lim_{x\rightarrow\infty}\frac{\sqrt{9x+1}}{\sqrt{x+1}}$$

Limits and continuity

### Find the limits: $$\lim_{x\rightarrow3}f(x),\ where\ f(x)=\frac{1}{2}x-7$$

Limits and continuity

### Find the limit and discuss the continuity of the function. $$\lim_{(x,y)\rightarrow(3,1)}(x^2-2y)$$

Limits and continuity

### Find the limit: $$\lim_{x\rightarrow\infty}\frac{3x^6-x^4+2x^2-3}{4x^5+2x^3-5}$$

Limits and continuity

### Use the method of your choice to evaluate the following limits. $$\lim_{(x,y)\rightarrow(0,1)}\frac{y\sin x}{x(y+1)}$$

Limits and continuity

### Evaluate the following limits. $$\lim_{(x,y)\rightarrow(e^2,4)}\ln\sqrt{xy}$$

Limits and continuity