NZ Level 8 (NZC) Level 3 (NCEA) [In development]
Applications of Limits

## Interactive practice questions

Find the asymptotes of $f\left(x\right)=\frac{e^x}{e^3-e^x}$f(x)=exe3ex.

Write the equations of all asymptotes on the same line, separated by commas.

Easy
Less than a minute

A particle with a rest mass $m_0$m0 and a velocity $v$v has a mass $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$m=m01v2c2, where $c$c is the speed of light.

What value does the mass approach as the velocity approaches the speed of light?

The total mass (in kilograms) of fish in a particular lake after $t$t years is modelled by $B\left(t\right)=\frac{4\times10^7}{1+6e^{-\frac{3t}{8}}}$B(t)=4×1071+6e3t8.

What will the total mass approach in the long run?

As each day passes, the water in a swimming pool is getting dirtier and dirtier. The concentration of dirt in the pool (in grams per litre) after $t$t days is modelled by $C\left(t\right)=\frac{4t}{90+t}$C(t)=4t90+t.

What is the concentration of dirt in the pool in the long run (as $t$t approaches infinity)?

### Outcomes

#### M8-10

Identify discontinuities and limits of functions

#### 91578

Apply differentiation methods in solving problems