Calculus

Topics within Calculus

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Practice Problems

Here are some problems to practice what you have learned! If you need a hint on any of them, there's a few for each problem.

1. $\displaystyle{\lim_{x\to 0} x+1 = }$

   Hint 1

   Remember that our ultimate goal is to substitute $x$ for $0$ when we can. Is there anything preventing us from computing $x+1$ when $x=0$?
   Show/Hide Solution

   We can plug in $x=0$ to the function, and see that $$\lim_{x\to 0} x+1=0+1=1$$.

2. $\displaystyle{\lim_{x \to 2} \frac{x^2 -4}{x-2}}= $

   Hint 1

   Always, when computing a limit, ask yourself: "Is there anything preventing me from just plugging in the value to the function?"
   Hint 2

   When you find what's preventing you from computing $\frac{x^2 -4}{x-2}$ when $x=2$, try to eliminate it.
   Show/Hide Solution

   We can't compute $\frac{x^2 -4}{x-2}$ when $x=2$ because then we would be dividing by $0$. However, remember that $$f(x)=\frac{x^2 -4}{x-2}= \frac{(x -2)(x+2)}{x-2}.$$ Remember that we are not finding $f(2)$, we are finding the limit of $f(x)$ as $x$ approaches $2$. So, we can divide the top and bottom of the fraction by $(x-2)$ because we know $(x-2)$ is not equal to 0, but rather very, very, close. Hence, our answer is $x+2=2+2=4$.

3. $\displaystyle{\lim_{x\to 0} \frac{1}{x}}= $

   Hint 1

   A limit, as you might remember, is a value that a function approaches as it nears the specified destination. Try graphing this function. What do you think the limit is?
   Hint 2

   Limits sometimes don't exist. For example, the limit of $g(x)=x$ as $x$ approaches $\infty$ doesn't exist, because there's really no way around not getting $\infty$ as your answer, which isn't an acceptable one. In this case, we say the limit doesn't exist. You may have heard the phrase in some video. Now you know what it means.
   Show/Hide Solution

   
   

   In the graph above, you see that the function just goes on and on and on... it doesn't stop! Who knows where it lands up at $0$?? Is it positive or negative? We say that the limit doesn't exist because as $x$ approaches $0$ from one side, it goes down, quite in disagreement with the other side. Additionally, there's nothing smart we can think of to get around it - at least I can't! Because the graph clearly tells us that the limit does not exist, this is what we live with ;). Note that this is a quite common case. Are there any other limits you can think of that don't exist?