How do I use properties of limits to evaluate a limit?

Answer 1

Let me first briefly go over the essential characteristics of limits so we can refer to them later.

Let #x_0 in mathbb{R}# and suppose that #lim_{x to x_0} f(x)# and #lim_{x to x_0} g(x)# exist and are finite. If #c in mathbb{R}# is a constant, the main properties of limits are: 1. #lim_{x to x_0} [cf(x)]=c[lim_{x to x_0} f(x)]# 2. #lim_{x to x_0} [f(x) pm g(x)]=[lim_{x to x_0} f(x)] pm [lim_{x to x_0} g(x)]# 3. #lim_{x to x_0} [f(x)*g(x)]=[lim_{x to x_0} f(x)] * [lim_{x to x_0} g(x)]# 4. If #lim_{x to x_0} g(x) ne 0# then we can state that #lim_{x to x_0} [f(x)/g(x)]=[lim_{x to x_0} f(x)] / [lim_{x to x_0} g(x)]#

These properties are very useful when computing limits (but take note of the conditions given at the beginning!). You can divide limits into smaller (and ideally easier) components.

Let's see a first example for property number 2: #lim_{x to 0} [e^x + log(x+1)]# It would be very annoying to solve this limit using the definition. But we can compute #lim_{x to 0} e^x=1# and #lim_{x to 0} log(x+1)=0# (I assume that the computation of these is done using the definition of limit, but I omit the calculation for space reasons). These two limits exist and are finite, so the conditions stated at the beginning hold. By the second property, the sum of these is the value of the initial limit: #lim_{x to 0} [e^x + log(x+1)]=1+0=1#.
Now an example where we focus on property number 4: #lim_{x to 0} {4x-3}/{x+1}#. If we compute #lim_{x to 0} (4x-3)=lim_{x to 0} (4x)-3=4 * lim_{x to 0} (x) -3=4*0-3=-3# (first we applied property number 2 and then property number 1 - all the conditions hold) and #lim_{x to 0} (x+1)=lim_{x to 0} (x) +1=0+1=1# (by property number 2), then we see that the conditions necessary to use the fourth property (existence and finiteness) hold. So we get #lim_{x to 0} {4x-3}/{x+1}={lim_{x to 0} (4x-3)}/{lim_{x to 0} (x+1)}={-3}/1=-3#
In these examples, the functions involved are continuous functions. Limits dealing with continuous functions are often easy to solve: if #f# is continuous in #x=x_0#, then #lim_{x to x_0} f(x)=f(x_0)# (this is often taken as the definition of continuity). Continuous functions are exactly the ones for which a simple substitution #x=x_0# solves the limit, and they don't let us see the importance of the properties listed at the top.
To better appreciate the properties listed above, let's see a case in which they aren't valid. To come up with these examples, it's necessary to involve non-continuous functions: #lim_{x to 0} [|x|/x * sin(x)]#. In this case, we can't evaluate #lim_{x to 0} |x|/x# and #lim_{x to 0} sin(x)# separately and then consider their product (trying to use property 3). In fact #lim_{x to 0} |x|/x# doesn't exist! How could we multiply a value that doesn't exist? A way to solve it is to rewrite it in this form: #lim_{x to 0} [|x| * sin(x)/x]#. Now we can use the third property, in fact #lim_{x to 0} |x|=0# and #lim_{x to 0} sin(x)/x=1# (this is one of the so-called notable limits), so they both exist and are finite! We can conclude that #lim_{x to 0} [|x| * sin(x)/x]=[lim_{x to 0} |x|] * [lim_{x to 0} sin(x)/x]=0 * 1 = 0#
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Answer 2

To use properties of limits to evaluate a limit, follow these steps:

  1. Direct Substitution: If the function is continuous at the point where the limit is being evaluated, simply substitute the value into the function.

  2. Algebraic Manipulation: Simplify the function algebraically by factoring, canceling common factors, or rationalizing the expression.

  3. Special Limits: Be aware of common limit results, such as ( \lim_{x \to 0} \frac{\sin x}{x} = 1 ) or ( \lim_{x \to \infty} \frac{1}{x^p} = 0 ), which can simplify the evaluation of certain limits.

  4. Limits of Sums, Differences, Products, and Quotients: Use the properties of limits to split the limit into smaller parts if it is in the form of a sum, difference, product, or quotient.

  5. Limit Laws: Apply limit laws, such as the sum law, difference law, product law, quotient law, and power law, to manipulate the limit expression.

  6. Squeeze Theorem: If the function is bounded between two other functions whose limits are known, then the limit of the bounded function is also known.

By applying these techniques and properties of limits, you can evaluate various types of limits efficiently and accurately.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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