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One Sided Limits
In a one-sided limit, the variable approaches the number from one side only. For example, the limit as x approaches 3 from the left, denoted x→3-, means the limit using only x-values to the left of 3, such as 2.9, 2.99, 2.999…. The limit as x approaches 3 from the right, denoted x→3+, means the limit using only x-values to the right of 3, such as 3.1, 3.01, 3.001….
Left and Right Limits
means the limit of f(x) as x → c, but with x < c.
means the limit of f(x) as x → c, but with x > c.
Two-Sided Limits
iff both one-sided limits exist and equal the same number L.
Ex A: Translate into words:
x → 3- x approaches 3 from the left
x → -3 x approaches negative 3 from both sides
x → -3- x approaches negative 3 from the left
x → 3- x approaches 3 from the left
x → -3 x approaches negative 3 from both sides
x → -3- x approaches negative 3 from the left
Ex B: Find the limits for the piecewise linear function
Ex B: Find the limits for the piecewise linear function
= d.n.e.
Ex C: Find the limit of a each function that has two variables.
Ex D: Find each limit by graphing the piecewise linear function.
Ex E: For each function graphed below, use the limit notation with ∞ and -∞ to describe its behavior as x approaches the vertical asymptote from the left, from the right, and from both sides.
Assignment
Part A: Translate into words.
Part B: Find each limit by substitution.
Part C: Find each limit.
Part D: Draw each graph by hand. Find the limit as x approaches 3 from the left and from the right. Find the two sided limit.
1.3 Notes & Homework
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1.3 Key
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