LimitsAndContinuity.pdf

2.1 THE LIMIT OF A FUNCTION AT A POINT 2.1 THE LIMIT OF A FUNCTION AT A POINT The Tangent Line Problem DEFINITION: The tangent line (or simply the tangent) to a curve at a given point is the straight line that ”just touches” the curve at that point, so that as it passes through the point of tangency, the tangent line has the same direction as the curve, and in this sense it is the best line approximation to the curve at that point. How do we find the equation of the tangent line to the graph of a function, say f(x), at the point (a,f(a))? . l . . . 0 rise f(x)−f(a) The slope of the line PQ is the difference quotient F(x) = run = x−a . 43 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY Example: Find the slope of the tangent line to the curve of f(x) = x at the point P(1,1). x < 1 F(x) x > 1 F(x) 0.9 1.1 0.99 1.01 0.999 1.001 0.9999 1.0001 0.99999 1.00001 → 1 − → → 1 + → DEFINITION: If f is a function, and L is a finite number, we say: “the limit of f(x), as x approaches a, equals L”, or “f(x) converges to L as x approaches a” and we write “ lim f(x) = L” or “f(x) → L as x → a” x→a if “the value of f(x) can be made arbitrarily close to L, for all x sufficiently near, but not equal to, a. 44 ▯2013 by Sophie Chrysostomou 2.1 THE LIMIT OF A FUNCTION AT A POINT sinx EXAMPLE: Investigate the behaviour of f(x) = x near a = 0. Note that f is not defined at a = 0. Find lim f(x). x→0 x < 0 f(x) x > 0 f(x) −0.1 0.1 −0.01 0.01 −0.001 0.001 → 0− → → 0+ → 2 HOMEWORK: Investigate the behaviour of f(x) = x near a = 2. 45 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY √ 9x − 36x + 36 Now let us investigate the behaviour of f(x − 2 near a = 2. Conclusions: x < 2 f(x) x > 2 f(x) → → → → 46 ▯2013 by Sophie Chrysostomou 2.2 ONE SIDED LIMITS 2.2 ONE SIDED LIMITS Consider the following function:  1  x + 1 if x ≥ 1 H(x) = 2  −1 if x < 1 with graph: 3 2 1 What is the lim H(x) ? x→1 -1 0 1 2 3 -1 . 47 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY DEFINITION: x→am+f(x) = L where L is a finite number is the right limit of f(x) as x approaches a from the right if the values of f(x) can be made arbitrarily close to L, for all x sufficiently close to a, but greater than a. HOMEWORK: lim f(x) = L where L is a finite number x→a − is the left limit of f(x) as x approaches a from the left 1.) If lx→a+(x) ▯= lx→a−(x), then we say that lix→a(x) does not exist. 2.) If lim f(x) = lim f(x) = L, and L is finite, then we say that x→a+ x→a− x→am f(x) exists and x→am f(x) = L. 48 ▯2013 by Sophie Chrysostomou 2.3 LIMIT LAWS 2.3 LIMIT LAWS 1. Constant Law If f is the constant function given by f(x) = c for all x ∈ domf, then lim f(x) = c. x→a 2.x→am x = a 3. Sum Law If lim f(x) = L and lim g(x) = M and L and M are finite x→a x→a numbers, then lim(f(x) + g(x)) = x→a 4. Product Law If lim f(x) = L and lim g(x) = M and L and M are x→a x→a finite numbers, then lim(f(x)g(x)) = x→a 49 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY 5. Quotient Law If lix→a(x) = L and lix→a(x) = M ▯= 0 and L and M are finite numbers, then lim(f(x)/g(x)) = x→a √ √ 6. Root Law lim nx = n a (Note: for n even we require a > 0.) x→a There is also a version of the root law that uses one-sided limits, which stat√s lim nx = 0 for n even. x→0+ 7. Basic Substitution Law Suppose that f is a polynomial or a rational function and a ∈ domf, then x→a f(x) = f(a). x − 1 x − 1 Examples: lim and lim x→1 x + 1 x→1 x − 1 50 ▯2013 by Sophie Chrysostomou 2.3 LIMIT LAWS 8. If I is an interval containing a and f(x) ≤ g(x) for all x ∈ I, except possibly at a, then: x→a f(x) ≤ lix→a(x) assuming both of these limits exist. 9. Squeeze Law (or Sandwich Theorem) Let I be an open interval containing a. Suppose that g(x) ≤ f(x) ≤ h(x) for all x ∈ I − {a}. If lim g(x) = L = lim h(x), then lim f(x) = L. x→a x→a x→a 1 EXAMPLE: Consider lim x sin 2 . x→0 x θ sinθ HOMEWORK: Show lim = 1 = lim . . θ→0 sinθ θ→0 θ (Hint: Show sinθcosθ < θ < tanθ) . θ tan θ sin θ 0 θ Circle of radius 1 51 ▯2013 by Sophie Chrysostomou LIMITS AND CONTINUITY 2.4 INFINITE LIMITS: VERTICAL ASYMPTOTES (f(x) → ±∞) 1 Consider lim 2. x→1 (1 − x) 16 14 12 10 8 6 4 2 -3 -2 -1 -2 1 2 3 4 5 Discussions: A) What is infinity? B) Is there a difference between i) lim f(x→a= ∞ and ii) lim f(x) dox→anot exist (or DNE as most are used to writing). DEFINITION: If I is an interval containing a, and f is defined for all x ∈ I, except possibly at a, x→a f(x) = ∞ means that the value of f(x) can be made arbitrarily large for all x suffi- ciently close to a, but not equal to a. HOMEWORK: Write similar definitions for any two of: x→a f(x) = −∞, lim+f(x) = ∞, lim+f(x) = −∞, x→a x→a lim− f(x) = ∞, lim −(x) = −∞ x→a x→a