Basic Limits in Calculus with The Math Sorcerer

Explore the intuitive idea of limits through reading limit notation, graph-based examples, one- and two-sided limits, holes and infinities, and how limits relate to derivatives and integrals.

Learn to compute basic limits by direct substitution, dropping the limit sign, and using factoring when needed, with examples like x approaches 1 and -2.

Learn to determine limits from a graph by evaluating left- and right-hand approaches, identify when they agree or differ, and recognize holes and solid dots that affect the limit value.

Plug in two into (T^2 − T)^5 to evaluate the limit as T approaches two, noting no fractions or square roots, and obtain 32 as the final answer.

Plug in p = 2 to evaluate the limit of 3p over (sqrt(4p+1) - 1) and simplify to 3, noting a real number result means no further tricks are needed.

Evaluate the limit as x approaches two for a linear function y = mx + b by substitution, giving 2m + b; recognize m as slope and b as intercept.

Show that a limit with a constant equals that constant, using three. Demonstrate that the limit is three from left or right, via the horizontal line y equals three.

Find the limit as x approaches c of b and see that it equals b, since the constant function yields the same value from left and right.

Evaluate the limit as x approaches 2 of (2 - x)/(x^2 - 4) by factoring the denominator as a difference of squares and canceling to obtain -1/4.

Learn to evaluate limits by substituting the variable as x approaches c, factoring out x minus c, and canceling terms to compute the limit, which resolves to minus one.

Evaluate a limit by applying the difference of squares with a minus b, cancel factors, and plug in near x = -1; compare factoring with multiplying, yielding -12.

Factor the numerator as a difference of squares: (5+h)^2 - 25 = h(10+h). Cancel h with the denominator to get the limit as h approaches zero, which equals 10.

Compute the limit as x approaches three by factoring the rational function, canceling the common factor, and substituting the value to obtain four.

evaluate the limit as x approaches 2 for a rational function by factoring the quadratic and canceling, then substitute 2 to get -5.

Factor the numerator as (x - c)(x - c) and cancel the common factor with the denominator. Substitute x = c to evaluate the limit, yielding 0.

Evaluate the limit as x approaches five by factoring and canceling to clear fractions, then substitute to get -1/36; learn using the LCD and alternative methods for quick limits.

Evaluate the limit as w approaches one by creating a common denominator, multiplying by w/w, and canceling to arrive at the limit equals -1.

Evaluate the limit as x approaches four by clearing the fraction, factoring, and canceling terms, then plug in four to obtain negative one over sixteen.

Evaluate a limit by rationalizing with the conjugate, using the a^2 - b^2 identity to cancel x - c, then plug in the value of c.

Learn to evaluate a limit involving a square root by rationalizing, using the conjugate and difference of squares to obtain the limit as h approaches zero.

Rationalize the expression to evaluate the limit as x approaches four, using the conjugate and the difference of squares to simplify and substitute.

Recognize the difference of cubes to factor the expression, cancel the common factor, and compute the limit as x approaches 2, yielding 12.

Evaluate the limit of f(x)=1/(x+3) as delta x approaches zero by substituting x+delta x, then simplify to relate to the derivative of the function, yielding -1/(x+3)^2.

Students learn to evaluate limit of (2/(x+Δx) - 2/x)/Δx by forming the common denominator, using the reciprocal of Δx, and taking the limit as Δx → 0 from the right.

Evaluate limits as delta x approaches zero; avoid zero in the denominator, distribute and cancel terms to reveal the limit equals five, the line y equals five.

Compute the limit as x approaches -8 by direct substitution, using x cubed to obtain -512.

Substitute x with 9 in 5x - 3 to evaluate the limit as x approaches 9. Shows the limit is 42 via direct substitution with no division by zero.

Apply the limit rule by plugging in 4 to simplify the expression, yielding 24 divided by sqrt(9) = 3, and arriving at the final limit value of 8.

Compute the limit as x approaches two for (x-2)/(x^2-4) by factoring the difference of squares and canceling, yielding 1/4; use L'Hôpital's rule on the zero over zero form to confirm.

Rewrite the numerator as -(x-2), factor the denominator as (x-2)(x+2), cancel the common factor, and evaluate the limit as x approaches 2 to get -1/4.

Learn to solve a limit by rewriting with the difference of squares, pulling out a negative one to cancel x-5, and evaluate at five to get -1/10.

Evaluate a limit by algebraic expansion and factoring, canceling h to simplify ((x+h)^2 - x^2)/h, then substitute h = 0 to obtain 2x.

Evaluate a limit by substituting and then factoring, using a difference of squares to cancel factors, and apply the limit as x approaches six to obtain seven over six.

Learn to evaluate a limit of a rational function as x approaches one by rewriting using difference of squares, canceling, and obtaining -1/2.

Evaluate the limit as x approaches zero for x over x^2+7x by factoring out x, canceling, and plugging in to get 1/7.

Solve a limit by factoring and canceling: as x approaches zero, (x^2+3x)/x simplifies to x+3, yielding the limit equal to 3.

Compute the limit as x approaches zero of (sqrt(x+3) - sqrt(3)) / x by rationalizing the numerator with the conjugate, simplifying via difference of squares, and obtaining 1/(2 sqrt(3)).

Evaluate the limit by rationalizing the numerator with the conjugate and applying the difference of squares. Plug in x approaching 1 to obtain 1/4.

Compute the limit as delta z approaches zero by expanding (z0 + Δz)^2, canceling and factoring Δz, then substituting to obtain 2 z0.

Determine the limit as u approaches infinity by focusing on leading terms in a polynomial over a polynomial, then optionally show work by dividing by u^4.

Evaluate the limit as x approaches negative infinity of x over sqrt(x^2+4) using two methods, intuitive absolute-value and algebraic, showing it equals -1.