Compute the following limits:
a) $$\displaystyle\lim_{x \to 0}{((2x+1)\cdot\dfrac{x^2-x}{x})}$$
b) $$\displaystyle\lim_{x \to -3}{x^3+x^2-x}$$
c) $$\displaystyle\lim_{x \to -\infty}{x^4+2x^2}$$
See development and solution
Development:
We are going to solve the following limits using the properties of limits that we already know:
a) $$\displaystyle\lim_{x \to 0}{((2x+1)\cdot\dfrac{x^2-x}{x})}=\lim_{x \to 0}{(2x+1)}\cdot\lim_{x \to 0}{\dfrac{x^2-x}{x}}=$$
$$\displaystyle=(2\cdot0+1)\cdot\lim_{x \to 0}{\dfrac{x(x-1)}{x}}=1\cdot\lim_{x \to 0}{x-1}=1\cdot(-1)=-1$$
b) $$\displaystyle\lim_{x \to -3}{x^3+x^2-x}=(-3)^3+(-3)^2-(-3)=-27+9+3=-15$$
c) $$\displaystyle\lim_{x \to -\infty}{x^4+2x^2}=(-\infty)^4+2\cdot(-\infty)^2=+\infty+2\infty=+\infty$$
Solution:
a) $$-1$$
b) $$-15$$
c) $$+\infty$$