Limits and Direct Substitution

In summary, a limit is a fundamental concept in calculus used to analyze the behavior of a function near a specific point. Direct substitution is a method for evaluating limits by plugging in a value and solving for the resulting expression. However, it can only be used when the resulting expression is a real number. To find a limit using direct substitution, the resulting expression must be undefined, other methods such as factoring or trigonometric identities must be used. Understanding limits and direct substitution is important because it allows for analysis of a function's behavior and can be used to find important values in mathematics and science.
  • #1
Qube
Gold Member
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Why wouldn't answer choice "C" be true as well?

[PLAIN]http://i.min.us/idPNiK.png

Homework Equations



I thought that you could use direct substitution on this limit.

The Attempt at a Solution



I did get answer choices A and B. For the limit to exist, both the left and right hand limits have to match up.
 
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  • #2
f(0) might not be defined or it may be that f(0)≠2. In other words, f(x) might have a removable discontinuity at x=2.
 

FAQ: Limits and Direct Substitution

What is a limit?

A limit is a fundamental concept in calculus that represents the value that a function approaches as its input approaches a specific value or as the input approaches infinity. It is denoted by the symbol "lim" and is used to analyze the behavior of a function near a certain point.

What is direct substitution?

Direct substitution is a method for evaluating limits by simply plugging in the given value into the function and solving for the resulting expression. It is only applicable when the given value does not cause the function to be undefined, such as division by zero or taking the square root of a negative number.

How do you use direct substitution to find a limit?

To use direct substitution to find a limit, simply plug in the given value into the function and solve for the resulting expression. If the resulting expression is a real number, then that is the limit. However, if the resulting expression is undefined, then you will need to use other methods, such as factoring or trigonometric identities, to find the limit.

Can you use direct substitution to evaluate all limits?

No, direct substitution can only be used to evaluate limits when the resulting expression is a real number. If the resulting expression is undefined, then other methods, such as factoring or trigonometric identities, must be used to find the limit.

Why is it important to understand limits and direct substitution?

Limits and direct substitution are important concepts in calculus because they allow us to analyze the behavior of a function near a certain point, even if the function is not defined at that point. They also allow us to find the values of limits, which can be used to find the slope of a tangent line, the rate of change of a function, and many other important applications in mathematics and science.

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