The epsilon-delta definition requires that for any arbitrarily small positive number epsilon, a corresponding delta can be found to ensure that f(x) remains within epsilon of L when x is within delta of c. This precise formulation is the foundation of understanding limits.

Since 2x + 1 is a linear and continuous function, the limit can be found by direct substitution. Substituting x = 3 yields 2(3) + 1 = 7.

The expression x² - 16 factors into (x - 4)(x + 4), allowing cancellation of (x - 4) with the denominator. After cancellation, substituting x = 4 into x + 4 yields 8.

A function is continuous at x = c if f(c) is defined, the limit as x approaches c exists, and that limit equals f(c). This ensures there is no jump, hole, or discontinuity at that point.

A constant function does not change its value regardless of x, so the limit as x approaches any point is simply k. This is one of the most basic properties of limits.

For rational functions where the highest power of x is the same in the numerator and denominator, the limit at infinity is determined by the ratio of the leading coefficients. Here, that ratio is 3/2.

This is a classic limit often used in the derivation of the derivative of the sine function. Direct substitution using small-angle approximations leads to a limit of 1.

Using the Taylor series expansion for cosine, 1 - cos x is approximately x²/2 for small values of x, so when divided by x² the limit evaluates to 1/2. This limit is frequently encountered in trigonometric limit applications.

As x approaches 0 from the right, the natural logarithm function decreases without bound. Therefore, the limit of ln x as x → 0❺ is -∞.

By rationalizing the expression and simplifying, the dominant terms yield an equivalent form that approaches 2 as x tends to infinity. This is a standard approach for limits involving square roots.

The difference of cubes in the numerator factors as (x - 1)(x² + x + 1), allowing cancellation of (x - 1) with the denominator. Substituting x = 1 into the simplified expression gives 1² + 1 + 1 = 3.

For small angles, tan x is approximately equal to x, leading the fraction tan x/x to approach 1. This limit is a direct consequence of small-angle approximations in trigonometry.

For x ≠ 1, the function simplifies to f(x) = x + 1 by canceling the common factor, yielding a limit of 2 as x → 1. To ensure continuity, f(1) must equal this limit, so k = 2.

Using the Taylor series for ln(1+x), ln(1+x) approximates x when x is close to zero, so the ratio ln(1+x)/x tends to 1. This result forms a foundational limit in calculus.

Since the highest powers of x in both numerator and denominator are the same, the limit is determined by the ratio of their leading coefficients. Here, that ratio is 2/3.

Expanding e^(2x) with its Taylor series yields 1 + 2x + 2x² + …; subtracting 1 and 2x leaves 2x² as the leading term. Dividing by x² then gives a limit of 2, resolving the indeterminate form.

Multiplying numerator and denominator by the conjugate simplifies the expression to 1/(√(x+4) + 2). Substituting x = 0 gives 1/(2 + 2), which equals 1/4. This technique effectively removes the radical from the numerator.

This limit is one of the classic definitions of the constant e. As x approaches 0, the expression (1+x)^(1/x) converges to e, a fundamental constant in mathematics.

Applying small-angle approximations, tan(3x) approximates 3x and sin(2x) approximates 2x for x near 0. Dividing these gives (3x)/(2x) = 3/2, which is the value of the limit.

Using the binomial expansion for square roots near 1, √(1 + sin x) approximates 1 + (sin x)/2 and √(1 - sin x) approximates 1 - (sin x)/2. Their difference is sin x, and dividing by x leads to the familiar limit sin x/x, which equals 1.