Calc 1 Flashcards

Horizontal asymptote: numerator degree < denominator degree

If the numerator has the smaller degree, the horizontal asymptote is y = 0.

Horizontal asymptote: numerator degree = denominator degree

If the degrees are equal, divide the leading coefficients: y = a/b.

Horizontal asymptote: numerator degree > denominator degree

If the numerator has the larger degree, there is no horizontal asymptote.

Degree of a polynomial

The highest exponent of x in the polynomial.

Limit with denominator approaching 0 and nonzero numerator

If the numerator approaches a nonzero number and the denominator approaches 0, check the signs to determine +∞ or -∞.

0/0 limit

If direct substitution gives 0/0, the limit is indeterminate and you need to simplify or use another method.

Nonzero/0 limit

If direct substitution gives a nonzero number divided by 0, check the left/right signs; the limit may be +∞, -∞, or DNE.

One-sided limit x → a+

The function approaches a from values greater than a.

One-sided limit x → a-

The function approaches a from values less than a.

Degree rule: top degree < bottom degree

Limit as x → ∞ is 0.

Degree rule: top degree = bottom degree

Limit as x → ∞ is the ratio of the leading coefficients.

Degree rule: top degree > bottom degree

The limit usually goes to +∞ or -∞; determine the sign from the leading terms.

Negative exponent rule

a^(-x) = 1/a^x.

Negative exponential limit

As x → ∞, a^(-x) → 0 when a > 1.

Limit: (sin 6x)/(3x) as x → ∞

0, because sin(6x) stays between -1 and 1 while 3x grows without bound.

Limit: (2x)/(tan x) as x → (π/2)-

0. Rewrite tan x as sin x/cos x, giving 2x cos x/sin x.

Limit: (3x)/(tan x) as x → (π/2)-

0.

Limit: (x² - 9)/(x² - 4x) as x → 4+

+∞. Factor the denominator as x(x - 4); from the right, x - 4 is positive and approaches 0.

Limit: (6x³ - 24x)/(9 - 2x²) as x → ∞

-∞. The leading terms give 6x³/(-2x²) = -3x.

Limit: (6x³ - 24x)/(9 - 2x²) as x → -∞

+∞. The leading terms give -3x, and -3(-∞) = +∞.

Limit: (6x² - 24x)/(9 - 2x²) as x → ∞

-3. The degrees are equal, so divide leading coefficients: 6/(-2) = -3.

Limit: (6x - 24)/(9 - 2x²) as x → ∞

0, because the denominator has the higher degree.

Limit: sqrt(5x⁴ - 3x)/(x² - 9) as x → ∞

√5. Factor x⁴ inside the square root and divide by x².

Limit: 3 + 2(7^(-x)) as x → ∞

3, because 7^(-x) → 0.

Limit: 5 + 4(3^(-x)) as x → ∞

5, because 3^(-x) → 0.

Limit: (4 + 2^(-x))/(3 + 5^(-x)) as x → ∞

4/3, because both negative-exponent terms approach 0.

Example: numerator degree 2, denominator degree 3

Limit as x → ∞ is 0.

Example: numerator degree 2, denominator degree 2

Divide the leading coefficients.

Example: numerator degree 3, denominator degree 2

Look at the leading terms; the result grows like a linear expression and may approach +∞ or -∞.