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Wize AP Calculus (BC) Textbook > Limits & Continuity

Limit Laws

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Limit Laws

We can compute limits more easily by using the following limit laws.

If lim⁡x→af(x)=L,lim⁡x→ag(x)=M \displaystyle\lim_{x\rightarrow a}f(x)=L, \displaystyle\lim_{x\rightarrow a}g(x)=M , and kk is a constant, then:
  1. lim⁡x→a[f(x)+g(x)]=L+M\displaystyle\lim_{x\to a}\left[f\left(x\right)+ g\left(x\right)\right]=L+M
  2. lim⁡x→a[f(x)−g(x)]=L−M\displaystyle\lim_{x\to a}\left[f\left(x\right)- g\left(x\right)\right]=L-M
  3. lim⁡x→a[f(x)g(x)]=LM\displaystyle\lim_{x\to a}[f\left(x\right)g\left(x\right)]=L M
  4. lim⁡x→a kf(x)=kL\displaystyle\lim_{x\to a}\ kf\left(x\right)=kL
  5. lim⁡x→a[f(x)g(x)]=LM,  (M≠0)\displaystyle\lim_{x\to a} \left[\frac{f\left(x\right)}{g\left(x\right)}\right] = \frac{L}{M},\ \ (M\neq0)
  6. lim⁡x→a[f(x)]mn=Lmn,  (L>0 if n is even, and L≠0 if m/n<0)\displaystyle\lim_{x\to a}\left[f\left(x\right)\right]^{\frac{m}{n}}=L^{\frac{m}{n}},\ \ (L>0\ \text{if }n\ \text{is even, and }L\neq0\ \text{if }m/n<0)
  7. If f(x)≤g(x)f(x)\le g(x) for every xx in an interval containing aa, then L≤ML\le M

Wize Tip
Direct Substitution
Always first try to substitute the value that xx approaches directly into the expression → if you get a number, that's the value of the limit!

Evaluate the limit:

lim⁡x→3[xx2−5+1]2\displaystyle\lim_{x\rightarrow3}\left[\frac{x}{x^2-5}+1\right]^2