Limit of functions
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Transcript of Limit of functions
LIMIT OF FUNCTIONS
One-sided limit, limit, existence of limit, limit at infinity, infinite limit
One-sided limit
Given a function defined by
Is the value y (real number) being approached by f(x) as x gets closer and closer to a from the right.
One-sided limit
Given a function defined by
Is the value y (real number) “being approached” by f(x) as x gets closer and closer to a from the left.
One-sided limit
Consider the function defined by
a. Evaluate b. Evaluate
One-sided limit
03 03
One-sided limit
392
x
xxf
x 392
x
xxf x 392
x
xxf
-3.1 -6.1 -2.9 -5.9 -3.01 -6.01 -2.99 -5.99 -3.001 -6.001 -2.999 -5.999 -3.0001 -6.0001 -2.9999 -5.9999
Table 2.1 Some numerical computations close to -3 from the left and right
One-sided limit
392
x
xxf
Theorem 2.1 Existence of LimitThe limit of a function exists if and only if the one-sided limits of the function are equal
One-sided limit
Given a function defined by
One-sided limit
Functions whose limit at a does not exist.Example 2.1
Evaluate , does it exist?
One-sided limit
a. Evaluate .b. Are they equal?
Conclusion: the limit does not exist
One-sided limit
Function whose limit does not exist at a.Example 2.4Evaluate where
One-sided limit
The limits do not point to a specific real number. Conclusion: limit does not exist as x approaches 3
Limit Theorems
Limit at infinity
Evaluate and , where
Limit at infinity
As x approaches positive infinity or negative infinity, the quotient approaches zero.
Limit at infinity
Evaluate where
Solution
Limit at Infinity and Horizontal AsymptoteObtain the horizontal asymptote of
Solution
Limit at infinity and horizontal asymptote
Horizontal Asymptote
Limit at infinity and asymptotes (horizontal and oblique)
Limit at infinity and asymptotes (horizontal and oblique)
Examples
Limit at infinity and asymptotes (horizontal and oblique) Perform long division
on
Limit of some trigonometric functions
Theorems
Limit of trigonometric functions
Limit of trigonometric functions
Limit of trigonometric functions
Limit of trigonometric functions