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3B MAS. 4. Functions. Limit of a Function. Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is obtained by moving along the curve from both sides of 'a' as x moves toward 'a'.
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3B MAS 4. Functions
Limit of a Function • Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is obtained by moving along the curve from both sides of 'a' as x moves toward 'a'. • The limiting value of f(x) as x gets closer and closer to 'a' is denoted by
Right/Left Hand Limits • As x moves towards 'a' from right (left) hand side, the limiting value of f(x) is denoted by
Limiting and Functional Value • If both sides limits are equal, • Otherwise, does not exist. • Note that may not equal to f(a)
y f(x) f(a) x a Limiting Value
a Example 1 Find the limit of the f(x) as x approaches a for the following functions. (a)
a Example 1 Find the limit of the f(x) as x approaches a for the following functions. (a) The limit does not exist as the function is not defined 'near' a.
a Example 1 (cont'd) (b)
a Example 1 (cont'd) (b) The limit does not exist as the left side limit is not the same as right side limit.
a f(a) Example 1 (cont'd) (c)
a f(a) Example 1 (cont'd) (c) The limit exists but it does not equal to f(a).
f(a) a Example 1 (cont'd) (d)
f(a) a Example 1 (cont'd) (d) The limit exists and it equals to f(a).
Evaluating Limits • If f(x) is not broken at 'a', use direct substitution to evaluate its limit as x approaches 'a' • Otherwise, find the left side and right side limits and check if they are equal.
Example 2 Evaluate the following limits if they exist. (a) f(x) = 2x – 5 as x 1 f(x) is not broken at x = 1, so use direct sub.
Example 2 (cont'd) (b) f(x) = ln x as x 0 f(0) is not defined. So consider limit from both sides. But f(x) is not defined for x < 0. So the limit does not exist.
Example 2 (c) f(x) = 1/(x – 2) as x 2 f(2) is not defined. So consider limit from both sides. Since the left side limit does not equal to the right side limit, the limit of the f(x) as x approaches 2 does not exist.
Example 2 (cont'd) (d) f(x) = (x – 1)/(x2 – 1) as x 1 f(x) = (x – 1)/(x + 1)(x – 1) = 1/(x + 1) 1/(x + 1) is not broken at x = 1, so use direct sub.
Limits to Infinity • If f(x) = x + c, f(x) as x (note that x is the dominant term) • If f(x) = 1/x, f(x) 0 as x • If f(x) = ax2 + bx + c, ax2 is the dominant term as x
Example 3 Find the limit of f(x) as x (if they exist) for:
Example 4 Find the following limits.
B C x O A r Consider the relationship between the areas OAC, sector OAC , andOAB
B C x O A r Example 4 (cont'd) Area of OAC = r2 sin x / 2Area of sector OAC = r2 x / 2Area of OAB = r2 tan x / 2
B C x O A r Area of OAC = r2 sin x / 2Area of sector OAC = r2 x / 2Area of OAB = r2 tan x / 2 So (size of areas) r2 sin x / 2 < r2 x / 2 < r2 tan x / 2 sin x < x < tan x1 < x / sin x < 1 / cos x1 > sin x / x > cos x Take limit as x 0 to get That means sin x x as x 0
Continuity • Graphically a graph is continuous at x = a if it is not broken (disconnected) at that point. • Algebraically the limit of the function from both sides of 'a' must equal to f(a).
a a f(a) a Example 5 The following functions are not continuous at x = a. Why?
f(a) a a Example 6 The following functions are continuous at x = a.
Example 7 • Determine if the given function is continuous at the given point. • f(x) = | x – 2 | at x = 2 • f(x) = x at x = 0 • f(x) = 1 / (x + 3) at x = -3
Example 8 Given that f(x) is continuous over the set of all real numbers, find the values of a and b.
Example 8 (cont'd) Only need to consider the junctions (x = -1 and x = 2)
Differentiability • Graphical approach: A function f(x) is said to be differentiable at x = a if there is no 'corner' or 'vertical tangency' at that point. • A function must be continuous (but not sufficient) in order that it may be differentiable at that point.
f(a) a Example 9 The following functions are not differentiable at x = a.(a) Corner at x = a
Example 9 (cont'd) (b) Vertical tangency at x = 1
Example 9 (cont'd) (c) Not continuous (not even defined) at x = -2
Example 10 The following functions are differentiable everywhere. (a)
Derivative of a Function • A function is differentiable at a point if it is continuous (not broken), smooth (no corner) and not vertical (no vertical tangency) at that point. • Its derivative is given by (First Principle)
Q f(x+h) P f(x) x x+h Differentiability (cont'd)
Differentiability (cont'd) • The gradient of PQ is given by • As Q moves closer and closer to P (i.e. as h tends to 0), the limiting value of the gradient of PQ (i.e. the derivative of f(x) at x) becomes the tangent at P.
Differentiability (cont'd) • The derivative of a function y = f(x) is denoted by • It also represents the rate of change of y with respect to x.
Example 11 • Find the gradient function of y = 2x2 using first principle. Find also the gradient at the point (3, 18). • Use the definition (first principle) to find the derivative of ln x and hence find the derivative of ex.
