Limits are the foundation of calculus, allowing us to analyze function behavior near specific points. They help us understand , rates of change, and function values at tricky spots.
Properties of limits simplify complex calculations by breaking them into manageable parts. We'll look at rules for basic operations, polynomials, powers, roots, and rational functions, as well as how to handle tricky situations like 0/0.
Properties of Limits
Limits of basic algebraic operations
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Top images from around the web for Limits of basic algebraic operations
How Do You Calculate a Limit Algebraically? – Math FAQ View original
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Finding Limits: Properties of Limits | Precalculus II View original
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Finding Limits: Numerical and Graphical Approaches · Precalculus View original
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states the of a sum equals the sum of the limits (limx→a(f(x)+g(x))=limx→af(x)+limx→ag(x))
Allows finding the limit of a sum by evaluating the limits of its individual components and adding the results
states the limit of a difference equals the difference of the limits (limx→a(f(x)−g(x))=limx→af(x)−limx→ag(x))
Enables finding the limit of a difference by evaluating the limits of its individual components and subtracting the results
states the limit of a product equals the product of the limits (limx→a(f(x)⋅g(x))=limx→af(x)⋅limx→ag(x))
Allows finding the limit of a product by evaluating the limits of its individual components and multiplying the results
states the limit of a constant multiple equals the constant multiple of the limit (limx→a(c⋅f(x))=c⋅limx→af(x), where c is a constant)
Enables finding the limit of a constant multiple by evaluating the limit of the function and multiplying it by the constant
Limits of polynomial functions
Polynomial functions are continuous everywhere, meaning the limit of a as x approaches a equals the value of the function at x=a
Continuity property simplifies the process of finding limits for polynomial functions
method finds limx→aP(x), where P(x) is a polynomial function, by evaluating P(a)
Plugging in the value of a directly into the polynomial function yields the limit value
Example: For P(x)=3x2−2x+1, limx→2P(x)=P(2)=3(2)2−2(2)+1=9
Limits with powers and roots
states the limit of a power equals the power of the limit (limx→a(f(x))n=(limx→af(x))n, where n is a real number)
Allows finding the limit of a power by evaluating the limit of the base function and raising it to the power
states the limit of an nth root equals the nth root of the limit (limx→anf(x)=nlimx→af(x), where n is a positive integer)
Enables finding the limit of an nth root by evaluating the limit of the radicand and taking the nth root of the result
states the limit of an exponential function equals the exponential of the limit (limx→abf(x)=blimx→af(x), where b>0 and b=1)
Allows finding the limit of an exponential function by evaluating the limit of the exponent and using it as the power of the base
Limits of rational functions
states the limit of a quotient equals the quotient of the limits, provided the limit of the denominator is not zero (limx→ag(x)f(x)=limx→ag(x)limx→af(x), where limx→ag(x)=0)
Allows finding the limit of a quotient by evaluating the limits of the numerator and denominator separately and dividing the results
occurs when both the numerator and denominator approach 0 as x approaches a, indicating the limit may exist but requires further investigation
or canceling common factors can simplify the expression and help determine the limit value
Indeterminate Form ∞∞ occurs when both the numerator and denominator approach ∞ or −∞ as x approaches a, indicating the limit may exist but requires further investigation
Dividing both the numerator and denominator by the highest power of x can help determine the limit value