Explain every equation on the image ---MaED 2322 (Calculus I) Worksheet 1
1. Use ε – δ definition to prove the following statements.
a. lim_(x→4) 1/x = 1/4
b. lim_(x→-2) (½ x - 1) = -3
c. lim_(x→-3) x² + 3x = 0
d. lim_(x→3) |x - 3| = 0
e. lim_(x→2) (x² - 4)/(x - 2) = 4
f. lim_(x→-3) (1 - 4x) = 13
g. lim_(x→1) (2x + 3) = 5
h. lim_(x→c) (mx - b) = mc + b where m ≠ 0
2. Evaluate the following limits.
a. lim_(x→2) √(2 - x)
b. lim_(x→4) (2x + 8√(x + 8))/(√(x - 2))
c. lim_(x→0) cos(x + π/2)/x
d. lim_(x→3) (2x + |x - 3|)
e. Let f(x) = { x² + 1 if x < 1
{ (x - 2)² if x ≥ 1
then find lim_(x→1⁻) f(x) and lim_(x→1⁺) f(x) does lim_(x→1) f(x) exist?
f. lim_(x→0) (3sin²x)/(x²)
g. lim_(x→0) (√3 + x - √x)/x
h. lim_(t→0) (1/(t√(1 + t)) - 1/t)
i. lim_(x→-6) (2x + 12)/|x + 6|
j. lim_(x→0) sin²x/(1 - cos x)
k. lim_(x→0) sinx/(³√x)
l. lim_(s→0) (√(1 + s) - 1)/s
3. Evaluate the following one sided limit.
a. lim_(x→-1⁺) (4x - 7x²)/(x + ½)
b. lim_(y→3⁻) -1/(√3 - y)
c. lim_(h→0⁺) (1/h - 1/√h)
4. Show that f is continuous on (-∞, ∞)
a. f(x) = { 1 - x if x ≤ 1
{ lnx if x > 1
b. f(x) = { sinx if x < π/4
{ cosx if x ≥ π/4
5. Find the numbers at which f is discontinuous. At which of these numbers if f continuous from the right, from the left, or neither?
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The epsilon-delta definition is the formal mathematical definition of a limit. It states that the limit of f(x) as x approaches a equals L if, for every epsilon greater than zero, there exists a delta greater than zero such that whenever x is within delta of a, f(x) is within epsilon of L. This definition provides the rigorous foundation for all limit calculations in calculus.
When evaluating limits, we have several standard techniques at our disposal. Direct substitution works when the function is continuous at the point. Factoring and cancellation help with indeterminate forms like zero over zero. The famous limit of sine x over x as x approaches zero equals one, which is fundamental in calculus and can be proven using the squeeze theorem.
One-sided limits examine the behavior of functions as we approach a point from only one direction. The left limit approaches from values less than the target, while the right limit approaches from values greater than the target. A two-sided limit exists only when both one-sided limits exist and are equal. When they differ, we have a jump discontinuity, as shown in the absolute value of x over x function.
Continuity requires three conditions: the function must be defined at the point, the limit must exist, and the limit must equal the function value. For piecewise functions, we must carefully check continuity at boundary points where the definition changes. In this example, both pieces approach the value 1 as x approaches 1, and the function value at x equals 1 is also 1, making the function continuous.
In summary, we have explored the fundamental concepts of limits and continuity that form the foundation of calculus. The epsilon-delta definition provides mathematical rigor, while various techniques help us evaluate limits in practice. Understanding one-sided limits and continuity is essential for analyzing function behavior, especially with piecewise functions. These concepts directly lead to derivatives and integrals, making them crucial for success in calculus.