Calculus 1 Made Easy

Explain the limit of a function with a hole at x equals 2, showing it approaches 5 from both sides using the graph and value tables.

Learn how to evaluate one-sided limits, distinguish left and right limits for piecewise functions, and determine when a two-sided limit exists as x approaches a point.

Master direct substitution to evaluate limits in calculus 1, using plug-in values for polynomials, radical expressions, and exponential forms, including one-sided limits.

Explain how to evaluate limits involving absolute value by analyzing the inner sign, using left and right limits, and the piecewise definition, with |x-2|/(x-2) yielding -1 as x approaches 2.

Explore the greatest integer (floor) function, its definition, and how limits from the left and right behave, highlighting existence only at non-integers.

Learn the dividing out technique for rational functions: factor the numerator and denominator, cancel a common factor, and substitute to determine the limit as x approaches a.

Solve a variety of calculus quiz questions on limits, left and right limits, and two-sided limits. Analyze the absolute value, piecewise functions, and the greatest integer function to assess continuity.

Master a quick trig limit shortcut: replace sin x and tan x with x as x approaches zero to simplify limits. Note exceptional additions or subtractions require other methods.

Investigate limits of trigonometric functions as x approaches zero, using sine, tangent, and cotangent relationships, substitution, and factoring to simplify expressions.

Explore how fractions with polynomials behave as the denominator approaches zero, producing positive infinity, negative infinity, or nonexistent limits; analyze left and right limits to determine the sign.

Divide by the highest power of x to find limits at infinity, keep only the highest-degree terms, ignore the rest, and determine the limit's sign as x goes to infinity.

Evaluate limits at infinity by focusing on dominant terms and using two methods: multiply brackets to obtain the leading term, or keep highest degree terms in each bracket.

Master limit problems by analyzing left and right limits, identifying infinite behavior, and applying dominant-term techniques—divide by the highest degree to evaluate rational functions as x grows.

Learn to solve limits by identifying dominant terms, dividing by the highest degree, and using absolute values for square roots as x tends to infinity or negative infinity.

examine limits at infinity of sign x, sine x, and cosine x as x approaches infinity or negative infinity; they do not exist due to oscillation between -1 and 1.

Explore limits of exponential functions as x grows or shrinks, distinguishing bases above one from bases between zero and one, with graphs and tables illustrating their end behavior.

Explore limits of rational and exponential functions using a shortcut method that keeps dominant terms, divides by the highest degree, and analyzes x approaching infinity and negative infinity.

Apply the squeeze theorem to a function between two others with equal limits, and use sine and cosine bounds to show that x^2 sin(1/x) and x cos(2/x) approach zero.

Use the squeeze theorem to evaluate limits: (x^2-1)cos(pi/(x-1)) as x→1+, x sin(1/x) as x→0+, and sqrt(x)e^{cos(pi/x)} as x→0+. Each limit equals 0.

Analyze continuity at a point by verifying the function value equals the limit, with left and right limits from piecewise definitions.

Find vertical asymptotes of rational functions by factoring, canceling common factors, and setting the denominator to zero. Cancellation may create a hole in the graph rather than an asymptote.

Determine horizontal asymptotes by taking limits as x approaches ±∞, keeping dominant terms, canceling others, and identifying y = L with examples like 2 and 3 from common rational functions.

Learn how to compute derivatives from the definition using limits and algebra, including conjugate techniques. See step-by-step examples for x^2, sqrt(x), and linear functions.

Apply the definition of derivative to compute limits and cancel factors for several functions, using conjugates and common denominators. Explore derivatives of rational, polynomial, and root expressions, including 0-point evaluation.

Learn to apply the power rule to differentiate expressions of the form x^n, bring the exponent down, reduce by one, and handle constants and sums.

Master the power rule for x^n, identify constants with zero derivative, and differentiate polynomials. Then apply these rules to evaluate derivatives at given points.

Apply the power rule to power functions, bring the exponent down, and reduce the power by one to differentiate with negative and fractional exponents.

Apply the product rule to differentiate the product of two functions, f and g, using f' g + f g', with examples demonstrating the derivative of their product.

Explore derivatives through power rule, product rule, and rewriting functions with negative exponents; apply them to evaluate derivatives at key values like -1, π, and 1.

Master the quotient rule for differentiating f over g. Learn the formula f'(x) g(x) - f(x) g'(x) over g(x)^2 and when rewriting the function helps simplify the task.

Use the quotient rule to differentiate (3x+4)/(2x+1) and compute y'(0) = -5. Differentiate f(x)=2x/(1+√x) and evaluate at x=1 to obtain f'(1)=3/4.

Learn the chain rule for differentiating composite functions by identifying the outside and inside functions, differentiating each, and multiplying their rates to obtain f'(g(x)).

Apply the chain rule to several examples by identifying inside and outside functions, using the power rule and quotient rule, and differentiating expressions like (x+1)/x^5 and sqrt(x).