How to solve limits with substitution

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Substitution is the simplest way to evaluate a limit, but it doesn’t always work

As we’ve seen in previous lessons, the simplest way to evaluate a limit is to substitute the value we’re approaching into the function.

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For instance, given the function $f(x)=x+1$ , finding the limit as $x\to5$ is as easy as substituting $x=5$ into $f(x)$ .

$\lim_{x\to5}(x+1)$

$5+1$

$6$

If $f(x)$ is an expression that contains only polynomials, roots, absolute values, exponentials, logarithms, trig or inverse trig functions, then we may be able to evaluate using substitution, and we’ll have

$\lim_{x\to a}f(x)=f(a)$

But if the function is undefined at $x=a$ , or if $x=a$ is the transition point between two pieces of a piecewise-defined function, then we can’t apply the substitution rule.

Nevertheless, when we evaluate a limit we should always try substitution first before any other technique, because it’s the easiest and fastest method. If substitution doesn’t work, then we can try evaluating the limit by a different method.