# Continuity and limits of functions

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Question 1

Figure 1.1 y = f(x) = (2x+4)2 - (2x - 4)2

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Apply the slope predictor formula to find the slope of the line tangent to Figure 1.1. Then write the equation of the line tangent to the graph of f at the point (3, f(3)).

Question 2

Find all points on the curve y = (x + 4)(x - 5) at which the tangent line is horizontal.

Question 3

Figure 3.1 100^/¯2 ft/sec

Figure 3.2 y = x - ( x 25 )2

Suppose that a projectile is fired at an angle of 45 degrees from the horizontal. Its initial position is the origin in the xy-plane, and its initial velocity is Figure 3.1. Then its trajectory will be the part of the parabola Figure 3.2 for which y ≥ 0. a) How far does the projectile travel (horizontally) before it hits the ground? b) What is the maximum height above the ground that the projectile attains?

Question 4

Evaluate limx => 16 ^/¯x-4 x-16

Question 5

Figure 5.1 f(x) = 4 ^/¯x+8

Given Figure 5.1, use the four-step process to find a slopepredictor function m(x). Then write an equation for the line tangent to the curve at the point x = 8.

Question 6

Use one-sided limits to find the limit or determine that the limit does not exist. limx => 4 16-x2 4-x

Question 7

Find the trigonometric limit: limx => 0 sin3x 2x

Question 8

Use the Squeeze Law of limits to find the limit. limx =>0 x2 sin2 10x

Question 9

Figure 9.1 h(x) = x-9 |x-9|

Given Figure 9.1, tell where h is continuous. (Give your answer in interval form.)

Question 10

Figure 10.1 f(x) = x-4 x2-16

Given Figure 10.1, find all points where f is not defined (and therefore not continuous). For each such point, tell whether or not the discontinuity is removable.

Question 11

Figure 11.1 f(x) = { c2-x2if x<0 ccosx if x>0

Given Figure 11.1, find a value for c so that f(x) is continuous for all x.

Question 12

Figure 12.1 Given Figure 12.1, tell where f is continuous. (Give your answer in interval notation.)

#### Solution Preview

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Question 1 Figure 1.1 y = f(x) = (2x+4)2 - (2x - 4)2

. Apply the slope predictor formula to find the slope of the line tangent to Figure 1.1. m =[ f(a+h) - f(a)]/hf(a+h) = [2(a+h) + 4]2 - [2(a+h) - 4]2 = 32(a+h)f(a) = [2a + 4]2 - [2a - 4]2 = 32am =[ f(a+h) - f(a)]/h = [32(a+h) - 32a]/h = 32Then write the equation of the line tangent to the graph of f at the point (3, f(3)). Slope is constant for all x.Slope at = 32Note: given function is (2x+4)2 - (2x-4)2 = 32 x which is a straight line with a slope of 32.Question 2 Find all points on the curve y = (x + 4)(x - 5) at which the tangent line is horizontal. y = x2 - x -20dy/dx = 2x - 1If the tangent line is horizontal, dy/dx = 0i.e. 2x - 1 = 0x = 0.5Tangent line is horizontal at x = 1/2Question 3 Figure 3.1 100^/¯2 ft/sec

Figure 3.2 y = x - ( x 25 )2

Suppose that a projectile is fired at an angle of 45 degrees from the ...

#### Solution Summary

Solutions to twelve problems involving functions, continuity, finding slope using predictor formula, tangent line to a curve, trajectory of a projectile, finding limits, finding limits using squeeze law and continuity of functions are provided. I have provided complete and detailed answer to all the questions. Solutions are in a 7-page word document.