Applied Calculus

Test on limits Limits by observation of graph Consider the graph of the function f (x) below: 1. Complete the right limit equation lim f (x) = x→−1+ 2. Complete the limit equation lim f (x) = x→−1 3. Write a left limit equation at x = 3 4. Write a right limit equation at x = 3 5. Does the limit at x = 3 exist? 6. Write a right limit equation at x = 1 7. Compute lim f (x) x→1− Polynomial functions 1. Compute the following limit lim x2 + 3x − 1 x→0 2. Compute the following limit lim (x2 − 1)(x + 1) x→1 3. Compute the following limit lim x2 − 10000x x→∞ Logarithmic functions 1. Consider the function f (x) = log (4x + 2) (a) Compute the domain of f (x). (b) Compute lim log (4x + 2) x→2 (c) Compute lim log (4x + 2) x→− 12 + (d) Compute lim log (4x + 2) x→∞ (e) Talk about the following limit equation: lim log(4x + 2) x→−10 Radicals 2. Consider the function √ f (x) = 3x + 1 (a) Compute the domain of f (x). (b) Compute limx→ 1 √ 3x + 1 3 (c) Compute limx→− 1 + √ 3x + 1 3 (d) Talk about the limit equation limx→−10 √ 3x + 1 Rational Functions 1. Consider the function f (x) = 3x + 1 4x − 2 (a) Compute the domain of f (x). (b) Compute lim 3x + 1 x→0 4x − 2 More on Rational Functions 1. Consider the function f (x) = x2 − 5x + 6 x−3 (a) Compute the domain of f (x). (b) Compute x2 − 5x − 6 x→3 x−3 lim 2. Consider the function f (x) = −1/(x + 2) + 1/2 x (a) Compute the domain of f (x). (b) Compute lim x→0 −1/(x + 2) + 1/2 x Other 1. Consider the function √ x−6 f (x) = x − 36 (a) Compute the domain of f (x). (b) Compute √ x−6 x→36 x − 36 lim 2. x4 − 1 x→∞ x5 + x2 − 1 lim 3. x2 − 1 x→∞ x2 + x − 1 lim Exponential Functions 1. Consider the function 2 f (x) = ex +3x−1 (a) Compute the domain of f (x). (b) Compute 2 lim ex +3x−1 x→2 2. Consider the function x2 −1 f (x) = e x−1 (a) Compute the domain of f (x). (b) Compute x2 −1 lim e x−1 x→1 Continuity Find the value of k such that the function ( x2 − x + 3, x ≤ 1 f (x) = 3x + k, x > 1 is continuos.