Calculus Basic Flashcards
Define the limit of f(x) as x approaches a.
The value L that f(x) gets arbitrarily close to when x is near a.
Evaluate lim(x→0) (sin x)/x.
1.
Evaluate lim(x→∞) 1/x.
0.
Evaluate lim(x→2) (x²−4)/(x−2) by factoring or direct substitution.
Use factoring: (x−2)(x+2)/(x−2)=x+2 → at x=2, result=4.
True/False: If f is continuous at x=a, then lim(x→a) f(x)=f(a).
True.
Find lim(x→∞) (2x²)/(x²+1).
Divide top/bottom by x² → limit=2/(1+0)=2.
lim(x→∞) (3x)/(x+4) in simplest form?
Divide by x: 3/(1+4/x)→3.
If f(x) has a removable discontinuity at x=a, how can it be fixed?
Redefine f(a)=lim(x→a) f(x) if that limit exists.
What does it mean if lim(x→a⁻) f(x)≠ lim(x→a⁺) f(x)?
f has a jump discontinuity at x=a (left/right limits differ).
State the Squeeze Theorem in short.
If g(x) ≤ f(x) ≤ h(x) and lim g(x)=lim h(x)=L, then lim f(x)=L.
Evaluate lim(x→∞) (5x+1)/(10x+3).
Divide top & bottom by x → (5+1/x)/(10+3/x)→5/10=1/2.
Evaluate lim(x→0) (1−cos x)/x² using known expansions.
Result=1/2 (using series or half-angle identity).
When is f(x) continuous at x=a?
If lim(x→a) f(x) exists, equals f(a), and f(a) is defined.
Find lim(x→∞) e^(−x).
0 (exponential decay).
Check continuity: f(x)=1/(x−1). Is it continuous at x=1?
No, it has a vertical asymptote at x=1.
lim(x→∞) (x²)/(e^x) → does it go to 0 or ∞?
0, because e^x grows faster than any polynomial.
Evaluate lim(x→π) (sin x)/(x−π) using L’Hôpital’s Rule or expansion.
Use L’Hôpital: derivative top cos x=−1, bottom=1 → limit=−1.
State L’Hôpital’s Rule condition in brief.
If limit is 0/0 or ∞/∞, we can take derivatives top and bottom.
If lim(x→2) f(x)=5, what is lim(x→2) [f(x)+3]?
5+3=8.
Define derivative of f at x=a: f’(a).
f’(a)=lim(h→0) [f(a+h)−f(a)]/h.
Derivative of f(x)=c (constant)?
0.
Derivative of f(x)=x^n?
nx^(n−1).
Derivative of f(x)=sin x?
cos x.
Derivative of f(x)=cos x?
−sin x.