Engineering Analysis

Calculus I

When to use direct substituion when solving for limits?

When a continuous function is present, or an intermediate is made into a continuous function.

If the limit of g(x) as it approaches a value of a is defined by the value b, and f(x) is continuous at the point b, what is the relationship between the limit of g(x) and the function f(x)?

lim x→a f(g(x)) = (f(lim x→a g(x)) = b) = f(b)

lim x→ 0 (sinx)/x = ?

1

lim x→ 0 (cosx - 1)/x = ?

0

lim x→ 0 (1- cosx)/x² = ?

½

lim x→ 0 (e^x - 1)/x = ?

1

lim x→ 0 ln(1 + x)/x = ?

1

lim x→ inf. (1 + 1/x)^x = ?

e

Rank the order of magnitude for x^p, b^x, lnx from highest to lowest

b^x → x^p → lnx

What is the lim x → inf. for a (low/high) order of magnitude

0

What is the lim x → inf. for a (same/same) order of magnitude

fraction of leading coefficients (coefficient numerator/coefficient denominator)

What is the lim x → inf. for a (high/low) order of magnitude

infinity

What is the lim x → +inf. 1/x^p ?

0

What are the five strategies for determining lim x → inf. ?

Divide by the highest power term, use logarithmic laws, multiply by the conjugate pair, let x = 1/t, and determine the limit based on the ratio of the orders of magnitude

What are the instances when limits do no exist?

Endpoints, Jump discontinuities, Oscillations, and Asymptotes

What is the domain and range of sin^-1x?

Dom: [-1,1] Ran: [-pi/2, pi/2] ***Inclusive Brackets***

What is the domain and range of cos^-1x?

Dom: [-1,1] Ran: [0,pi]

What is the domain and range of tan^-1x?

Dom: All real numbers Ran: (-pi/2,pi/2) ***Exclusive brackets***

What is the lim x→ + inf. (1/x^p) where p is some integer?

0

What is the lim x→ 0 (1/x^p) where p is an even number

+ inf.

What is the lim x → 0 (1/x^p) where p is an odd number

- inf.