LIMITS AND DERIVATIVES
Frequently Asked Questions
A limit describes the value that a function \(f(x)\) approaches as \(x\) approaches a particular number, written as \(\lim_{x\to a} f(x)\).
It means that the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently close to \(a\), but not necessarily equal to \(a\).
No, the limit depends on the behavior of the function near the point, not necessarily on the value of \(f(a)\).
The left-hand limit is \(\lim_{x\to a^-} f(x)\), where \(x\) approaches \(a\) from values less than \(a\).
The right-hand limit is \(\lim_{x\to a^+} f(x)\), where \(x\) approaches \(a\) from values greater than \(a\).
A limit exists at \(x=a\) if both left-hand and right-hand limits exist and are equal.
An infinite limit occurs when \(f(x)\) increases or decreases without bound as \(x\) approaches a value, written as \(\lim_{x\to a} f(x)=\infty\).
For a constant function \(f(x)=c\), \(\lim_{x\to a} c = c\) for any real number \(a\).
For \(f(x)=x\), \(\lim_{x\to a} x = a\).
If \(\lim_{x\to a} f(x)=L\) and \(\lim_{x\to a} g(x)=M\), then \(\lim_{x\to a} [f(x)+g(x)]=L+M\).
\(\lim_{x\to a} [f(x)-g(x)] = L-M\), provided the individual limits exist.
For a constant \(k\), \(\lim_{x\to a} kf(x)=k\lim_{x\to a} f(x)=kL\).
\(\lim_{x\to a} [f(x)g(x)] = LM\), if both limits exist.
\(\lim_{x\to a} \frac{f(x)}{g(x)}=\frac{L}{M}\), provided \(M\neq 0\).
The limit of a polynomial at \(x=a\) is found by direct substitution of \(x=a\).