Find the slope and intercept of a linear function from two limit values

Expert intro: A linear function behaves nicely under limits, so the given limits are really just equations in disguise. Once you recognize that, the system falls apart quickly.

Detailed walkthrough

Because $f(x)=mx+b$ is linear, it is continuous everywhere. That means

$\lim_{x\to 1}f(x)=f(1)=-2$

and

$\lim_{x\to -1}f(x)=f(-1)=4$

So we get the system

$m+b=-2$

$-m+b=4$

Add the equations:

$2b=2\Rightarrow b=1$

Substitute back:

$m+1=-2\Rightarrow m=-3$

Answer

$\boxed{m=-3,\ b=1}$

If you want, you can also check:

$f(x)=-3x+1$

Then $f(1)=-2$ and $f(-1)=4$, which matches both limits.

💡 Pitfall guide

Don’t overthink the word limit here. For a polynomial or linear function, the limit at a point is just the function value at that point. Another frequent error is writing the equations in the wrong order and solving for the signs incorrectly. A quick substitution check at the end prevents that.

🔄 Real-world variant

If the function were quadratic instead, two limit values would not be enough to determine all coefficients unless one coefficient were already known. For a linear function, though, any two distinct input-output conditions give a unique line. If one of the limits were changed, you would just solve a different 2×2 system in the same way.

🔍 Related terms

continuity of linear functions, system of equations, limit equals function value