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# 5 2 limit-ofa_function

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### Transcript

• 1. Limit of a Function
• 2.
• The concept of a limit is central to the study of calculus.
• Differentiation and integration are each based on the notion of a limit.
• Limit of f ( x ) as x approaches a – the value that f ( x ) approaches as x gets closer to a
• In symbols:
• 3.
• Observe the values of f ( x ) = x 2 + 1 as x approaches 3:
• We say that .
x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4 f ( x ) 5 7.25 9.41 9.94 9.994 10 10.006 10.06 10.61 13.25 17
• 4.
• 5.
• Given . Find the limit of f ( x ) as x approaches 3. .
• We still say that .
x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4 f ( x ) 5 7.25 9.41 9.94 9.994 15 10.006 10.06 10.61 13.25 17
• 6.
• Given . Find the limit of f ( x ) as x approaches 2. .
x 1 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 3 f ( x ) 3 4 4.8 4.98 4.998 4 4.004 4.04 4.41 6.25 9
• 7.
• The limit of the function f exists only if the left-hand and right-hand limits both exist and are equal.
• exists if and only if
• 8. Limit Theorems
• 9.
• L1. The limit of a constant function is the constant.
• Examples.
• 10.
• L2. If f is a polynomial function, then
• Example.
• 11.
• L3. The limit of a sum is the sum of limits.
• Example.
• 12.
• L4. The limit of a constant times a function is the constant times the limit of the function.
• Example.
• 13.
• L5. The limit of a product is the product of the limits.
• Example.
• 14.
• L6. The limit of a quotient is the quotient of the limits.
• Example.
• 15.
• L7. The limit of a function raised to an exponent is the limit of the function raised to that exponent.
• Example.
• 16.
• Example. Find .
• 17.
• Example. Find .
• 18.
• Example. Given Find . .
• 19.
• 1.a Find .
• 20.
• 1.d Find .
• 21.
• 2.c Find .
• 22.
• 3.a Find .
• 23.
• 3.c Find .
• 24.
• 4.b Find .
• 25.
• 5.a Suppose and . Find . .
• 26.
• 7. Given . Find . .