Lecture 10: Limits Involving Infinity & Asymptotes of a Graph

Limits

limf(x)=L(finite)\lim f(x) = L (finite) 

x±x\rightarrow \pm\infty 

Remember \rightarrow lim1xp=0\lim \frac{1}{x^p} = 0 where p=±p = \pm

x±x\rightarrow \pm\infty 

limk=k\lim k=k where K = constant

x±x\rightarrow \pm\infty 


Horizontal Asymptote

The line y=L is H.A of the curve y=f(x)y=f(x) if \rightarrow

limf(x)=L\lim f(x) = L or limf(x)=L\lim f(x) = L

x+x→+∞  x x→-∞


Remember this!

  1. limx3x2+x7=0\lim \frac{x-3}{x2} + x-7 = 0

xx→∞ 

2. limx2+5x7x3= \lim \frac{ x2 +5x-7}{x-3} = ∞ no. HA

xx→∞ 

3. limx2+5x74x2+3=14\lim \frac{x2 +5x-7}{4x2 + 3} = \frac{1}{4} therefore HA @ y=14y= \frac{1}{4}

xx→∞

4. limex=0\lim e^x = 0 & limex=\lim e^{-x} = ∞

x x→∞

5. limex=\lim e^x = ∞ & limex=0\lim e^{-x} = 0

x x→∞

6. lim1x2=\lim \frac{1}{x2}= ∞ therefore limDNE\lim DNE & lim1x=\lim \frac{1}{x} = -∞ , therefore limDNE\lim DNE

x0 x→0 x0x→0

Vertical Asymptote

The line x=ax=a  is a VA of the curve y=f(x) y= f(x) if

limf(x)=±\lim f(x) = \pm∞ or lim=±\lim = \pm∞

xa+x→a+  xx→-∞








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