The Limit of a Function

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Calculating Limits Using the Limit Laws

Continuity

Infinite Limits; Vertical Asymptotes

Limits at Infinity; Horizontal Asymptotes

Gottfried Wilhelm von Leibniz Gottfried Leibniz was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.

Tangents The word tangent is derived from the Latin word tangens, which means “touching.” Thus, a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can his idea be made precise? For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once. For more complicated curves this definition is inadequate.

Sir Isaac Newton, (4 January 1643 – 31 March 1727) was an English physicist, mathematician, astronomer, alchemist, and natural philosopher who is generally regarded as one of the greatest scientists in history. Newton wrote the Philosophiae Naturalis Principia Mathematica, in which he described universal gravitation and the three laws of motion, laying the groundwork for classical mechanics. By deriving Kepler's laws of planetary motion from this system, he was the first to show that the motion of objects on Earth and of celestial bodies are governed by the same set of natural laws.

Instantaneous Velocity; Average Velocity If you watch the speedometer of a car as you travel in city traffic, you see that the needle doesn’t stay still for very long; that is, the velocity of the car is not constant. We assume from watching the speedometer that the car has a definite velocity at each moment, but how is the “instantaneous” velocity defined? In general, suppose an object moves along a straight line according to an equation of motion, where is the displacement (directed distance) of the object from the origin at time. The function that describes the motion is called the position function of the object. In the time interval from to the change in position is. The average velocity over this time interval is

Now suppose we compute the average velocities over shorter and shorter time intervals. In other words, we let approach. We define the velocity or instantaneous velocity at time to be the limit of these average velocities: This means that the velocity at time is equal to the slope of the tangent line at.

The Derivative as a Function

Differentiable Functions

Match the graph of each function in (a)–(d) with the graph of its derivative in I–IV. Give reasons for your choices.