How do I use properties of limits to evaluate a limit?
Let me first briefly go over the essential characteristics of limits so we can refer to them later.
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.
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To use properties of limits to evaluate a limit, follow these steps:

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

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

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.

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.

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

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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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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