BackEstimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let f(x)=(x² − 1)/(|x| − 1).
b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−1.
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→3 (3x − 7) = 2
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→9 √(x − 5) = 2
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 √(4 − x) = 2
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→1 f(x) = 1 if f(x) = {x², x ≠ 1
2, x = 1
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→1 1/x = 1
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx → √3 1/x² = 1/3
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→−3 (x² − 9) / (x + 3) = −6
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 x² sin (1/x) = 0
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Slope of a Curve at a Point
In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.
y=x³−3x²+4, P(2,0)
Slope of a Curve at a Point
In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.
y=x³, P(2,8)
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
f. limx→0 f(x) = 0
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
j. limx→2− f(x) = 2
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
b. limx→2 f(x) does not exist
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
e. limx→1+ f(x) = 1