In this text we"ll just introduce a couple of simple methods for evaluating limits and show you part examples. The more formal methods of finding limits will it is in left for calculus.

You are watching: Find the limit of the function by using direct substitution.

A limit of a role at a specific x-value go not count on the worth of the role for that x. Therefore one an approach for assessing a border is analyzing a role for countless x-values an extremely close come the desired x. For example, f (x) = 3x. What is f (x)? Let"s find the values of f at some x-values close to 4. F (3.99) = 11.97, f (3.9999) = 11.9997, f (4.01) = 12.03, andf (4.0001) = 12.0003. From this, that is safe to speak that as x philosophies 4, f (x) ideologies 12. The is to say, f (x) = 12.

The an approach of analyzing a function for countless values the x near the desired value is fairly tedious. For details functions, a lot easier technique works: straight substitution. In the trouble above, we could have simply evaluated f (4) = 12, and had ours limit v one calculation. Since a limit at a given value that x does not count on the value of the role at the x-value, direct substitution is a shortcut that does not constantly work. Often a role is unknown at the preferred x-value, and in part functions, the worth of f (a)≠

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f (x). So straight substitution is a method that should be tried through most attributes (because that is so quick and easy come do) but constantly double-checked. It tends to work for the borders of polynomials and also trigonometric functions, but is much less reliable for functions which space undefined at specific values of x.

The various other simple method for detect a limit involves straight substitution, yet requires an ext creativity. If straight substitution is attempted, but the role is undefined because that the offered value of x, algebraic techniques for simplifying a duty may be supplied to findan expression the the role for i beg your pardon the worth of the function at the wanted x is defined. Then straight substitution can be offered to uncover the limit. Together algebraic techniques encompass factoring and also rationalizing the denominator, to surname a few. However a function is manipulated for this reason that direct substitution might work, the price still need to be confirm by one of two people looking at the graph of the function or examining the function for x-values near the desired value. Now we"ll look in ~ a couple of examples of limits.

What is ?

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Figure %: f (x) = By straight substitution and also verification native the graph, = -
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.

What is ?

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Figure %: f (x) = direct substitution doesn"t work, since f is unknown at x = 1. By facotring the denominator into (x + 1)(x - 1), though, the (x - 1) term cancels ~ above the top and also bottom, and we space left examining
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. By direct substitution, the border is
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.

Consider the duty f (x) = xforx f (x) = x + 1forx≥ 0. What is f (x), what is f (x), and also what is f (x)?

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Figure %: f (x) = x because that x , f (x) = x + 1 for x≥ 0The one-sided border from the left is 0. This we have the right to tell both from straight substitution and also by examining the graph. Making use of the same techniques, we discover the one-sided limit from the ideal is 1. Through the rules of the nonexistent limit, f (x) does no exist, because is f (x)≠f (x).

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Consider the duty f (x) = xforallx≠3, f (x) = 2forx = 3. What is

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f (x)?
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Figure %: f (x) = x for every x≠3, f (x) = 2 because that x = 3Direct substitution provides the limit at 2, but an ext careful inspection of the graph and the values surrounding x = 3 show that in reality the limit of f in ~ x = 3 is 3. This is a prime instance of exactly how the worth of a role at x has no impact on the limit of that role at x.