There are many techniques for finding limits that apply in various conditions. It's important to know all these techniques, but it's also important to know when to apply which technique.
Here's a handy dandy flow chart to help you calculate limits.
Key point #1: Direct substitution is the go-to method. Use other methods only when this fails, otherwise you're probably doing more work than you need to be. For example, it would be extra work to factor an expression into a simpler form if direct substitution would have worked without the factoring.
Key point #2: There's a big difference between getting b/0b/0 and 0/00/0 (where b0b\neq 0). When you get b/0b/0, that indicates that the limit doesn't exist and is probably unbounded (an asymptote). In contrast, when you get 0/00/0, that indicates that you don't have enough information to determine whether or not the limit exists, which is why it's called the indeterminate form. If you wind up here, you've got more work to do, which is where the bottom half of the flow chart comes into play.
Note: There's a powerful method for finding limits called l'Hôpital's rule, which you'll learn later on. It's not covered here because we haven't learned about derivatives yet.

Practice with direct substitution

problem 1
g(x)=x3x+53g(x)=\dfrac{x-3}{\sqrt{x+5}-3}
We want to find limx4g(x)\displaystyle\lim_{x\to4}g(x).
What happens when we use direct substitution?
Choose 1 answer:
Choose 1 answer:
Problem 2
h(x)=1cos(x)2sin2(x)h(x)=\dfrac{1-\cos(x)}{2\sin^2(x)}
We want to find limx0h(x)\displaystyle\lim_{x\to 0}h(x).
What happens when we use direct substitution?
Choose 1 answer:
Choose 1 answer:

Practice with the indeterminate form

Problem 3
Justin tried to find limx1x+1x2+3x+2\displaystyle\lim_{x\to-1}\dfrac{x+1}{x^2+3x+2}.
Using direct substitution, he got 00\dfrac00.
For Justin's next step, which method would apply?
Choose 1 answer:
Choose 1 answer:
Problem 4
Catherine tried to find limx34x+284x+3\displaystyle\lim_{x\to -3}\dfrac{\sqrt{4x+28}-4}{x+3}.
Using direct substitution, she got 00\dfrac00.
For Catherine's next step, which method would apply?
Choose 1 answer:
Choose 1 answer:

Putting it all together

Problem 5
Jill's teacher gave her a flow chart (below) and asked her to find limx5f(x)\displaystyle\lim_{x\to 5}f(x) for f(x)=x225x210x+25f(x)=\dfrac{x^2-25}{x^2-10x+25}.
Drag the cards below to show Jill's path to finding the limit.
A. Direct substitution
B. Asymptote
C. Limit found
D. Indeterminate form
E. Factoring
F. Conjugates
G. Trig identities
H. Approximation
Problem 6
Fenyang's teacher gave him a flow chart (below) and asked him to find limx3f(x)\displaystyle\lim_{x\to 3}f(x) for f(x)=2x51x3f(x)=\dfrac{\sqrt{2x-5}-1}{x-3}.
Drag the cards below to show Fenyang's path to finding the limit.
A. Direct substitution
B. Asymptote
C. Limit found
D. Indeterminate form
E. Factoring
F. Conjugates
G. Trig identities
H. Approximation
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