Overview:

# Part 2: Limits

### Limits

Roughly speaking, a limit of a function is the value that function approaches as the input into the function approaches some value.

Consider the function $$\text{f}(x)=\dfrac{1}{x}$$. As $$x$$ gets larger, the value of $$\text{f}(x)$$ decreases, getting closer and closer to 0, but never reaching or dropping below 0. We write that $$\displaystyle \lim_{x \to \infty} f(x) = 0$$, and say this as follows: “the limit as $$x$$ tends to infinity of $$\text{f}(x)$$ is 0.”

This applet has four examples. In each case, the aim is to understand what happens to the value of $$\text{f}(x)$$ as $$x$$ increases. Use the yellow “+1” button to repeatedly increase the value of $$x$$ by 1. Once you’ve done this a few times, consider the values of $$\text{f}(100)\text{, f}(1000)\text{, f}(1,000,000)$$ etc. to get a sense of the limits reached by these functions.

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Compare the above examples with the function $$\text{f}(x)=x^2$$. In this case, you can hopefully see that as $$x$$ increases, so does the value of $$\text{f}(x)$$. There is no limit in this case.

##### Exercise
Now have a go at finding the following:
1. $$\displaystyle \lim_{x \to \infty} \frac{2}{x}$$
2. $$\displaystyle \lim_{x \to \infty} \frac{3}{x}$$
3. $$\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}+7\right)$$
4. $$\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}-7\right)$$
5. $$\displaystyle \lim_{x \to \infty} \frac{1}{x^2}$$
6. $$\displaystyle \lim_{x \to \infty} 2^{-x}$$
7. $$\displaystyle \lim_{x \to \infty} \left(2^{-x}+5\right)$$
8. $$\displaystyle \lim_{n \to \infty} \frac{n}{n+1}$$
9. $$\displaystyle \lim_{n \to \infty} \frac{n}{n+2}$$
10. $$\displaystyle \lim_{n \to \infty} \frac{3n}{n+2}$$
11. $$\displaystyle \lim_{n \to \infty} \frac{3n}{4n+2}$$
12. $$\displaystyle \lim_{t \to 0} \frac{2t^2+7t}{t}$$
1. $$\displaystyle \lim_{x \to \infty} \frac{2}{x}=0$$
2. $$\displaystyle \lim_{x \to \infty} \frac{3}{x}=0$$
3. $$\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}+7\right)=7$$
4. $$\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}-7\right)=-7$$
5. $$\displaystyle \lim_{x \to \infty} \frac{1}{x^2}=0$$
6. $$\displaystyle \lim_{x \to \infty} 2^{-x}=0$$
7. $$\displaystyle \lim_{x \to \infty} \left(2^{-x}+5\right)=5$$
8. $$\displaystyle \lim_{n \to \infty} \frac{n}{n+1}=1$$
9. $$\displaystyle \lim_{n \to \infty} \frac{n}{n+2}=1$$
10. $$\displaystyle \lim_{n \to \infty} \frac{3n}{n+2}=3$$
11. $$\displaystyle \lim_{n \to \infty} \frac{3n}{4n+2}=\frac{3}{4}$$
12. $$\displaystyle \lim_{t \to 0} \frac{2t^2+7t}{t}=7$$