Question

I need help quick plz!

How do you solve this? I have the answer key from my teacher which he provided but I’m very confused on how to solve it. I’m stuck on the conjugating part. The answer is 1/6. However, I need the steps to get that.

Answer 1

Multiply the numerator and denominator by
`√(t^2+9)+3`. The motivation for this is to exploit the difference of squares identity,

`a^2-b^2=(a-b)(a+b)`

In this case,
`a=√(t^2+9)` and
`b=3`. So we have

`(√(t^2+9)-3)/(t^2)=(√(t^2+9)-3)/(t^2)\cdot(√(t^2+9)+3)/(√(t^2+9)+3)`

`(√(t^2+9)-3)/(t^2)=((√(t^2+9))^2-3^2)/(t^2(√(t^2+9)+3))`

`(√(t^2+9)-3)/(t^2)=((t^2+9)-9)/(t^2(√(t^2+9)+3))`

`(√(t^2+9)-3)/(t^2)=(t^2)/(t^2(√(t^2+9)+3))`

`(√(t^2+9)-3)/(t^2)=\frac1{√(t^2+9)+3}`

Now the expression in the limit is continuous at
`t=0`, so the limit can be computed by substitution:

`\displaystyle\lim_(t\to0)(√(t^2+9)-3)/(t^2)=\lim_(t\to0)\frac1{√(t^2+9)+3}=\frac1{√(9)+3}=\boxed{\frac16}`