# Determine the following limits. lim_{xrightarrow1}frac{1-x^2}{(x-1)^2}

Question
Limits and continuity
Determine the following limits.
$$\lim_{x\rightarrow1}\frac{1-x^2}{(x-1)^2}$$

2021-03-03
Given information:
$$\lim_{x\rightarrow1}\frac{1-x^2}{(x-1)^2}$$
To determine the limits first simplify,
$$\lim_{x\rightarrow1}\frac{1-x^2}{(x-1)^2}=\lim_{x\rightarrow1}(\frac{(1-x)(1+x)}{(x-1)^2}$$
$$=\lim_{x\rightarrow1}(\frac{(1+x)}{(x-1)})$$
$$=\lim_{x\rightarrow1}(1+x)\cdot\lim_{x\rightarrow1}(\frac{1}{(x-1)})$$
$$=(1+1)\cdot(\frac{1}{1-1})$$
$$=2(\frac{1}{0})$$
$$=\infty$$

### Relevant Questions

Evaluate the following limits or determine that they do not exist.
$$\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{x^2z-3x^2-y^2z+3y^2}{xz-3x-yz+3y}$$
Find the function The following limits represent the slope of a curve y=f(x) at the point {a,f(a)} Determine a possible function f and number a, then calculate the limit
$$\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}$$
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
$$\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}$$
Use the method of your choice to evaluate the following limits.
$$\lim_{(x,y)\rightarrow(2,0)}\frac{1-\cos y}{xy^2}$$
Use Taylor series to evaluate the following limits.
$$\lim_{x\rightarrow0}\frac{\sec x-\cos x-x^2}{x^4} \ (Hint: \text{The Maclaurin series for sec x is }1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+...)$$
Use the method of your choice to evaluate the following limits.
$$\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{2x^2-xy-y^2}$$
$$\lim_{(x,y)\rightarrow(0,\pi/2)}\frac{1-\cos xy}{4x^2y^3}$$
$$\lim_{x\rightarrow0}\frac{\sqrt{1+2x}-1-x}{x^2}$$
$$\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{yz-xy-xz-x^2}{yz+xy+xz-y^2}$$
$$\lim_{x\rightarrow0}\frac{\tanh^{-1}x}{\tan(\pi x/2)}$$