Limit Basics
What is a limit?
The value a function approaches as the input approaches a point.
What is continuity at a point?
The function value equals the limit at that point with no gaps or jumps.
How do you find the limit of (3x + 1) as x approaches 2?
Substitute x = 2; limit is 3 times 2 plus 1 equals 7.
What is the derivative of a function at a point?
The instantaneous rate of change or slope of the tangent line at that point.
What does a positive derivative indicate about a function?
The function is increasing at that point.
What does it mean if the limit of f(x) as x approaches a equals L?
As x gets closer to a, f(x) gets closer to L.
Name one type of discontinuity.
Jump discontinuity, removable discontinuity, or infinite discontinuity.
How do you handle limits that give 0 divided by 0?
Use algebraic simplification or factoring to eliminate the indeterminate form.
State the power rule for derivatives.
The derivative of x to the n is n times x to the (n minus 1).
How do you find critical points?
Set the derivative equal to zero or find where it does not exist and solve.
What is the difference between one-sided limits?
Left-hand limit approaches from values less than a; right-hand limit from values greater than a.
What must be true for f(x) to be continuous at x = a?
f(a) is defined, the limit of f(x) as x approaches a exists, and both are equal.
Find the limit of (x squared minus 1) divided by (x minus 1) as x approaches 1.
Factor numerator: (x minus 1)(x plus 1) over (x minus 1) equals x plus 1; substitute x = 1, limit is 2.
What derivative rule would you use to find the derivative of x squared times sine x?
The product rule.
How can derivatives help find local maxima and minima?
Use the first or second derivative test on critical points.
Why are limits important in calculus?
They help define instantaneous rates of change and the derivative.
How can continuity affect limit calculations?
If f is continuous at a, then the limit of f(x) as x approaches a equals f(a).
Evaluate the limit of sine x divided by x as x approaches 0.
The limit is 1 (a standard trigonometric limit).
Write the chain rule formula in words.
The derivative of a composite function f of g of x equals the derivative of f at g(x) times the derivative of g at x.
What does the second derivative tell you?
It tells if the function is concave up (second derivative positive) or concave down (second derivative negative).
What does it mean if a limit does not exist?
The function approaches different values from left and right or grows without bound.
Can a function be continuous on an interval but not differentiable there? Give an example.
Yes; for example, f(x) = absolute value of x is continuous everywhere but not differentiable at x = 0.
Explain how to find limits approaching infinity for rational functions.
Compare degrees of numerator and denominator; if degrees are equal, limit is the ratio of leading coefficients.
Find the derivative of (x squared plus 1) divided by x using the quotient rule.
The derivative is [(2x)(x) minus (x squared plus 1)(1)] divided by x squared, which simplifies to (x squared minus 1) divided by x squared.
Explain how derivatives are used in optimization problems.
Find where the derivative is zero to identify maximum or minimum values for practical problems.